Thermo-Electrically Pumped Semiconductor Light Emitting Diodes

Thermo-Electrically Pumped

Semiconductor Light Emitting Diodes

by

Parthiban Santhanam A0 ESB.S., University of California at Berkeley (2006) MASSCHU$ETfS INS-1 1

OFTECHNOLOGYS.M., Massachusetts Institute of Technology (2009)

APR 10 201Submitted to the Department of

Electrical Engineering and Computer Science LIBRARIESin partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Electrical Engineering

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

February 2014

© Massachusetts Institute of Technology 2014. All rights reserved.

A uthor ................. ....................... ...........Department of

Electrical Engineering and Computer ScienceJanuary)4, 2014

C ertified by .................................... .........Rajeev J. R-am

Professor of Electrical EngineeringThesis Supervisor

n)

Accepted by .................................. .'r 1 .........Le A. Kolodziej ski

Chair, Department Committee on Graduate Theses

%, - I .

Thermo-Electrically Pumped

Semiconductor Light Emitting Diodes

by

Parthiban Santhanam

Submitted to the Department ofElectrical Engineering and Computer Science

on January 14, 2014, in partial fulfillment of therequirements for the degree of

Doctor of Philosophy in Electrical Engineering

Abstract

Thermo-electric heat exchange in semiconductor light emitting diodes (LEDs) allowsthese devices to emit optical power in excess of the electrical power used to drivethem, with the remaining power drawn from ambient heat. In the language of semi-classical electron transport, the electrons and holes within the device absorb latticephonons as they diffuse from their respective contacts into the LED's active region.There they undergo bimolecular radiative recombination and release energy in theform of photons. In essence the LED is acting as a thermodynamic heat pump oper-ating between the cold reservoir of the lattice and the hot reservoir of the outgoingphoton field.

In this thesis we report the first known experimental evidence of an LED behaving as aheat pump. Heat pumping behavior is observed in mid-infrared LEDs at sub-thermalforward bias voltages, where electrical-to-optical power conversion at arbitrarily highefficiency is possible in the limit of low optical output power. In this regime, thebasic thermal physics of an LED differs from that seen at conventional higher voltageoperating points. We construct a theoretical model for entropy transport in an LEDheat pump and examine its consequences both theoretically and experimentally. Weuse these results to propose a new design for an LED capable of very high efficiencypower conversion at power densities closer to the limit imposed by the Second Lawof Thermodynamics. We then explore the potential application of these thermo-photonic heat pumps as extremely efficient sources for low-power communication andhigh-temperature absorption spectroscopy.

Thesis Supervisor: Rajeev J. RamTitle: Professor of Electrical Engineering

3

Acknowledgments

The work described in this thesis represents the collective efforts of a number of

people. I'd like to take a minute to recognize a few of them.

I feel I should begin where everything I've done has, with my family. Over the

course of my formal education, I have slowly come to realize the incredible impact

that the attitudes of my parents toward knowledge and learning have had on me.

As long as I can remember, they have woven the process of learning with the other

joys of life, and have thereby contributed to the quality of my life immeasurably. I

remember vividly the emphasis my father placed on the fundamentals as he taught

me math on weekends. I believe there is a direct connection from those experiences to

my approach to research and for that he deserves my thanks. In more recent times, I

have looked to them for help and guidance more often than I could have anticipated.

In response they have been more understanding than I thought possible and were

always generous with their unwavering love and support. My sister and her family

have been the closest family members within driving distance for several years now.

They have served as a constant reminder that the often myopic mindset of graduate

school is not all that life has to offer. In a very concrete sense, I could not have

reached the point I'm at without them. I only hope I can return the favor someday.

In good faith I cannot omit the countless friends, roommates, classmates, and

nontrivial combinations thereof who have supported my growth through conversa-

tion, cohabitation, cooperation, commemoration, and occasionally commiseration.

My former roommates Shawn Henderson and Matt McFall, both of whom I have

been lucky enough to call friends for more than half my life, have been two of my

closest companions and I hope they will continue to be in the coming phases of life.

My friend Rachel VanCott has been a constant presence in a time of fluctuation; Mike

Rosenberg has shared many of the interests I have carried since childhood and helped

in the dissipation of my cravings to watch and play sports. Laura Dargus has always

had an open seat, a free minute, and plenty of empathy, and I won't soon forget the

chats we've had in her office. David Hucul and Nabil Iqbal have been remarkable

5

catalysts for getting out and doing fun stuff. Donny Winston's zest for life has left

me with some unbelievable stories and a friend whom I can always count on. The

Cookie Monday regulars, my Intramural sports teammates, my fellow Wichita trans-

plants, the WAKA Kickballers, the many easygoing RLE admins, the VP crew, and

my Ashdown/Sid-Pac friends have all given me countless happy memories and played

a real role in making my twenties what they have been.

Several professional relationships deserve mention here. First and foremost, my

work would not have been possible without the generous funding I have received from

the EECS Department, the Office of Naval Research, the NDSEG Fellowship Program,

and Weatherford Int'l. Of the many MIT faculty members whose classes I hope never

to forget, I was fortunate to have on my thesis committee four of the professors I've

most admired. Professor Mehran Kardar and Professor Lizhong Zheng, from whom I

took Statistical Mechanics and Information Theory respectively, rank highly on that

list. I was delighted to have them on my thesis committee, through which I was able

to get feedback from points of view outside the semiconductor device community. I

was also lucky to have Professor Vladimir Bulovic, whose enthusiasm for academic

research has luminesced brightly as a research advisor and as the Director of MTL, on

my committee; his interest in applying our thinking to organic LEDs was instrumental

in clarifying the assumptions underlying our theoretical framework. In a similar vein,

my discussions with collaborators including Prof. Ali Shakouri, Dr. Je-Hyeong Bahk,

Dr. Mona Zebarjadi, Prof. Boris Matveev, Dr. Jess Ford, Dr. Ligong Wang were

necessary parts of the work described in this thesis.

Many of my fellow students have also contributed significantly. From Prof. Qing

Hu's group, David, Ivan, Qi, Wilt, and Sushil were always ready to discuss new ideas,

lend equipment and teaching time, and generally foster an enjoyable and productive

atmosphere for research. Prof. Ben Williams, Dr. Alan Lee, and Dr. Tom Liptay

were senior figures when I first came to MIT, and I probably took away more advice

from each of them than they know. I owe a special thanks to Prof. Dave Weld for

the time he took from his postdoc and first year as junior faculty at UCSB to provide

feedback and walk me through my first article submission to Physical Review Letters.

6

It was an important point in graduate school for me, and someday I hope to emulate

his genuine and patient encouragement.

As part of Rajeev Ram's Physical Optics and Electronics Group, several of my

labmates have been so many things to me- role models, coffee buddies, friends, sources

of advice, and sanity checks. I have shared so much of my experience in the last five

years with Dodd Gray- both professionally and personally. He was the yin to my

yang during our early work with low-biased LEDs and was an absolute rock of moral

support in the years before our work was published. Duanni Huang's persistence in

building the communication experiment was admirable and working with him pro-

vided me with important lessons in mentorship. More recently, Bill Herrington and

Priyanka Chatterjee have brought the lab to life with their fresh perspectives and

I look forward to working with them going forward. When Karan Mehta came to

our group, we immediately bonded over our interest in physics and the conversations

we shared during walks and over coffee have shaped many of the physical pictures I

rely on daily. Jason Orcutt was the consummate professional in lab, or at least as

much as a graduate student can be without losing their street cred. Over the years

I have often asked myself "What Would Jason Do?" and I continue to emulate him

in many ways. I will remember Reja Amatya for her seemingly effortless work ethic

and her choice to pursue the kind of research project that makes the world a better

place. Kevin Lee's humor and high spirits brightened the atmosphere in the group,

and his amazing nose bubble video will live on in the lab's lore. Tauhid Zaman was

a one-man minority in his appreciation of the ten-page handouts on Second Quanti-

zation that I may never live down, and from what I remember, he was never bashful

about anything really. Johanna Chong raised my opinion of the MIT undergraduate

experience and has always been a good friend. Shireen Goh's organizational skills

remain a model for me, and I wish her the best in her new life in Singapore. Evelyn

Kapusta was a hoot. I only hope that I can retain my "cloud person" status forever.

I'd also like to thank William Loh for his technical perspective and willingness to sit

down and explain things with patience.

During my grad school years my research advisor Prof. Rajeev J. Ram had an

7

enormous influence on me. As a teacher, mentor, role model, and finally a colleague,

I have been the beneficiary of his attitudes toward many things in research and

in life. During the first week of graduate school, I attended a welcome lecture by

some senior academic official at which the ideal of an advisor's role was likened to

"academic fatherhood." Aside from the unnecessarily gendered word choice, I felt that

description fit my goal as well. I had been told by many of my fellow grad students

that such a relationship was overly idealistic and these days impossible. Perhaps it

is because I was fortunate enough to work with Rajeev, but in retrospect this view

strikes me as cynical, and I consider myself lucky to have avoided it.

I still remember many of the conversations I've had with Rajeev. He shared

his views on the importance of role models, how to find the right research project,

and why so many people struggle with their twenties these days. One of the more

memorable methods he employed was to tell a Buddhist parable. Here I'd like to

approximate returning the favor.

There once was an American living in Japan, who while hiking in a forest came

across an old man outside his secluded home. As he was keen to practice his Japanese,

he began a conversation. The old man said he was a martial arts instructor and offered

to teach the American a lesson in karate. The American accepted the offer and worked

hard to be a good student. At the end of the lesson, the old man offered to teach

him again the next day, and the American accepted the gracious offer. That night,

the American went back to the city and told some of his American friends about his

new sensei and one of them asked to tag along. The next day two Americans came to

the old man, and he taught them both. Again at the end of the lesson he offered to

teach them again the next day. For weeks this pattern continued, with the American

students increasing in number until the sensei had a full class. One day at the end

of class, the students got together and decided they should offer to pay the old man

for teaching them. They approached him with their offer, but the old man declined.

When the students insisted that his teaching was so good that they felt like they

should be paying for it, the old man replied: "if I decided to charge you, you couldn't

afford me."

8

In the same way, the lessons Rajeev has taught me are valuable, but since he has

so much to give the world, so is his time. From my perspective, the dedication he

shows toward his graduate students seems beyond compensation. He must do it for

better reasons. My plan is to pay it forward. Thanks again, Rajeev, for your time

and energy.

9

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10

Contents

1 Background 15

1.1 LED Efficiency and Heat ...... ................ ....... 16

1.2 The LED as a Thermodynamic Heat Engine . . . . . . . . . . . . . . 21

1.3 Previous Work Toward Unity Efficiency . . . . . . . . . . . . . . . . . 27

1.4 Efficient Communication with a Photonic Heat Pump . . . . . . . . . 30

1.5 Potential Practical Applications . . . . . . . . . . . . . . . . . . . . . 35

1.5.1 Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 35

1.5.2 Lighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.5.3 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . 40

1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2 LEDs as Heat Pumps 43

2.1 Electron Transport and Entropy Flow in LEDs . . . . . . . . . . . . . 43

2.1.1 Current Continuity . . . . . . . . . . . . . . . . . . . . . . . . 44

2.1.2 Quasi-Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 46

2.1.3 Thermally-Assisted Injection . . . . . . . . . . . . . . . . . . . 47

2.1.4 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.1.5 Continuity of Entropy Flux . . . . . . . . . . . . . . . . . . . 52

2.2 The Heat Pump Picture . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3 LEDs in the Low-Bias Regime . . . . . . . . . . . . . . . . . . . . . . 58

2.4 Carnot-Efficient LEDs and Real LEDs . . . . . . . . . . . . . . . . . 64

2.4.1 Carnot Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4.2 Non-Ideality of Existing LEDs . . . . . . . . . . . . . . . . . . 66

11

2.4.3 The Power-Efficiency Trade-Off . . . . . . . . . . . . . . . . . 67

2.5 Design of LEDs for Heat Pumping . . . . . . . . . . . . . . . . . . . . 69

2.6 Circuits are Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 81

3 Experiments on Existing Emitters 83

3.1 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.1.1 Current-Biased Lock-In Technique . . . . . . . . . . . . . . . . 85

3.1.2 Temperature Control . . . . . . . . . . . . . . . . . . . . . . . 88

3.1.3 Thermal Shock of LED Packaging . . . . . . . . . . . . . . . . 90

3.1.4 Optical Design . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.2 Demonstration of r/ > 1: A = 2.5pm . . . . . . . . . . . . . . . . . . . 94

3.3 High Power Attempt: A = 4.7pm . . . . . . . . . . . . . . . . . . . . 99

3.4 Lower Emitter Temperatures: A = 3.4pm . . . . . . . . . . . . . . . . 101

3.4.1 Exclusion of Emissivity Modulation . . . . . . . . . . . . . . . 101

3.4.2 Unity Efficiency at Room Temperature . . . . . . . . . . . . . 108

3.4.3 Does Voltage Determine Brightness? . . . . . . . . . . . . . . 114


4 Communication with a Thermo-Photonic Heat Pump 119

4.1 Power Measurements as Slow Communication . . . . . . . . . . . . . 119

4.1.1 Sample Calculation . . . . . . . . . . . . . . . . . . . . . . . . 120

4.1.2 Extrapolation to Low Power . . . . . . . . . . . . . . . . . . . 125

4.1.3 Extrapolation to Carnot-efficient LEDs . . . . . . . . . . . . . 129

4.2 Limits of Energy-Efficient Communication with a Heat Pump . . . . 131

4.2.1 The Entropy Trade-Off . . . . . . . . . . . . . . . . . . . . . . 131

4.2.2 Calculation of the kBT ln(2) Limit . . . . . . . . . . . . . . . . 133

4.3 A Thermo-Photonic Link . . . . . . . . . . . . . . . . . . . . . . . . . 142


12

5 High-Temperature mid-IR Absorption Spectroscopy

5.1 M otivation . . . . . . . .. . . . . . . . . . . . . . . .

5.2 Mapping Spectroscopy onto Communication . . . . .

5.3 High-Temperature Sources for Spectroscopy.....

5.4 High-Temperature Infrared Photo-Detection . . . . .

5.5 High-Temperature Emitter-Detector Compensation .

5.6 Summary and Conclusions . . . . . . . . . . . . . . .

6 Conclusions and Future Work

6.1 Thesis Summary and Conclusions . . . . . . . . . . .

6.2 Further Scientific Questions . . . . . . . . . . . . . .

6.2.1 Entropy and Information in Photons . . . . .

6.2.2 Entropy and Information in Electrons . . . . .

6.3 Further Applied Directions . . . . . . . . . . . . . . .

6.4 Engineering Toward Second Law Bounds . . . . . . .

A Entropy and Temperature of Light

B Maximum Efficiency at 1 Sun

References

13

155

. . . . 156

. . . . 157

. . . . 160

. . . . 164

. . . . 171

. . . . 176

179

180

184

184

189

192

195

201

213

227


14

Chapter 1

Background

In the last two decades opto-electronic devices such as diode lasers, photo-voltaics,

and light-emitting diodes (LEDs) have been developed with improved capabilities at

drastically reduced costs. As a result, widespread use of these devices is no longer ex-

clusive to the traditional applications that have historically driven their development

[1]. Beyond their historical use as indicator lights, LEDs have been widely adopted

for displays [2], sources in spectroscopic applications [3, 4, 5], automotive applica-

tions [6], outdoor lighting [7], and increasingly the markets for indoor commercial

and residential lighting [8].

In this thesis we consider LEDs as thermodynamic heat pumps. In § 1.1 we estab-

lish the basic thermal physics of traditional LED operation. In § 1.2 we demonstrate

that this behavior stands in contradiction to what should be expected from a heat

pump. In § 1.3 we review the literature on LED heat-pumping in anticipation of

presenting its experimental observation in Chapter 3. Some theoretical and practical

consequences of the heat-pumping regime are motivated in § 1.4 and § 1.5 respec-

tively, before they are analyzed more fully in later chapters. In § 1.6 we provide a

short outline of the thesis as a whole.

15

1.1 LED Efficiency and Heat

As the roles of light-emitting diodes expand, the variety of operating conditions they

are subjected to is broadening and the demands on their performance are rising.

Their performance in high-temperature environments remains a ubiquitous challenge,

as suggested by Figure 1-1. The efficiency of LEDs depends strongly on the thermal

environment in which they operate. To explain the physical origin of this dependence

in both the traditional and heat-pumping regimes, we begin by briefly reviewing the

physics of a conventional double-hetero-junction LED. A simplified band diagram of

such a device has been adapted from [9] and appears in Figure 1-2.

The wall-plug efficiency q of an LED is defined as the ratio of emitted optical

power L to the supplied electrical power IV. Since each electron that passes through

the device has some probability of emitting a photon of energy hw, the efficiency 77

may be decomposed in terms of this probability (here denoted iEQE):

(hwyn7EQEqV

() (Rradiative) active

qV (RSRH)active + (Rradiative)active ± (RAuger)active

Here (lw) is the average energy of the emitted photons, q is the magnitude of the

electron's charge, and V is the applied voltage; the external quantum efficiency EQE

is the ratio of the rate at which photons exit the device to the rate at which electrons

pass through it as current. As shown, EQE may be further decomposed into the

efficiency with which generated photons are extracted from the device (qextract), the

efficiency with which injected electrons from the cathode and injected holes from

the anode fall into the narrow-gap active region and remain confined there until

recombination (7inject), and the fraction of that active-region recombination which is

radiative.

Here we have also assumed that all recombination events outside the active region

do not contribute to useful light (these events are accounted for by Tinject < 1) and

that all recombination events inside the active region are of one of three types [1, 14]:

16

0

1 A=1.9mQ

Xz4.7im0-

X=2.1pm

X=3.4pm

50 100 150 200 250 300 350 400 450 500Temperature (K)

Figure 1-1: Electrical-to-optical power conversion efficiency at typical operating cur-rents versus temperature for several modern LEDs emitting at various wavelengths.The green dashed line shows the performance of an InGaN LED (h ~ 2eV) from2011 [7]; the blue squares and the black dashed line are from two near-infrared In-GaAsSb LEDs (hw 600meV) from 2006 [10] and 2009 [11] respectively; the blackcircles are from a mid-infrared InAs LED (hw 350meV) from 2002 [12]; the solid redline is from a long-wavelength InAsSb LED (hw 250meV) from 2009 [13]. In spiteof the range of wavelengths and the variety of material systems in which they werefabricated, for each of these devices, the efficiency clearly decreases with increasingtemperature.

17

0M

a) .

Ep tico

- - F

EFp

Position

nE

Figure 1-2: Simplified band diagram of a conventional double-hetero-junction LED.The solid lines indicate the edges of the conduction and valence bands (labeled Ecand Ev respectively). The dashed lines indicate the Fermi level in the metal contactsand the electron and hole quasi-Fermi levels in the semiconductor regions (labeledEFn and EFp respectively). The wavy line denotes an exiting photon. The soliddouble-line between the diamonds represents the imaginary boundary which is crossedexactly once for each quantum of charge that flows as current. Charge may cross thedouble-line by either thermionic emission of minority carriers (i.e. carrier leakage) oractive region recombination events. The double-hetero-junction structure is generallydesigned to confine carriers to minimize leakage and thereby increase overall efficiency.Figure adapted from [9].

18

trap-assisted Shockley-Read-Hall (SRH) [15, 16], radiative, or Auger [17]. Their rates

(typically expressed in units of cm-3 s- 1 ), are denoted above as RSRH, Rradiative, and

RAuger respectively; the symbol (- )atjve denotes an average over the active region

volume.

Both the injection efficiency Tinject and the non-radiative Auger recombination rate

RAuger are strongly dependent on temperature. Together they are responsible for the

decreased efficiency of LEDs with operating temperature [17].

In Figure 1-2, the solid double-line between the diamonds represents the imaginary

boundary which is crossed exactly once for each quantum of charge that flows as

current. Charge may cross the double-line in one of 3 ways: (1) by the net thermionic

emission of an electron over the p-side conduction band hetero-barrier at left, (2) by

a recombination event in the active region in the middle, or (3) by the net thermionic

emission of a hole over the n-side valence band hetero-barrier at right. The double-

hetero-junction structure is generally designed to confine carriers so that current of

type (2) dominates over (1) and (3), leading to high minject. Since the parasitic leakage

processes (1) and (3) are thermionic emission processes over finite barriers, their rates

in typical operating regimes are exponentially dependent on temperature.

The rate of non-radiative Auger recombination is also exponentially dependent

on temperature in a similar way. The Auger process may be visualized as the time-

reversed version of impact ionization. In impact ionization, a high-energy electron

collides with an electron in the valence band to promote it to the conduction band.

The final states of both electrons must also have the same total momentum as the

initial electron states. In Auger recombination, an electron recombines with a hole of

different momentum (i.e. a non-vertical inter-band transition), and gives this energy

to another free electron which subsequently relaxes non-radiatively. Because of the

momentum-difference required of the original electron-hole pair, states very near the

band-edge of direct-gap semiconductors are not sufficient. Instead, the Auger process

requires the carriers undergoing recombination to inhabit excited initial states. This

requirement causes the temperature-dependence of RAuger, as it is dependent on the

presence of carriers with kinetic energy above some threshold energy which depends

19

on the bandstructure.

In summary, elevated temperatures traditionally cause LEDs to be less efficient, as

excited carriers undergo more non-radiative Auger recombination and the quality of

carrier confinement is reduced. Degraded device performance may be seen empirically

in Figure 1-1 to hold across virtually the entire range of commercially-available LED

emission wavelengths.

Moreover, even at room temperature the inefficiency of the LED itself leads to heat

generation which may further degrade performance. State-of-the-art visible LEDs

fabricated from InGaN achieve high internal quantum efficiency at low power density,

but at higher current density the portion of input power which is not emitted as

light results in substantial self-heating. This heating contributes to the so-called

"efficiency droop" [18, 191, thereby reducing the potential for energy savings from

solid-state lighting [7]. It also decreases bulb lifetime, thereby increasing amortized

capital and installation costs of LED lighting solutions [7].

LEDs designed to emit photons in the spectroscopically-valuable mid- and far-

infrared wavelength ranges also face major thermal challenges. In the mid-infrared

(A=2-8pm), state-of-the-art LEDs are at most 1-3% efficient [12, 11, 13, 20]. The re-

maining 97-99% of the electrical drive power results in self-heating; the consequences

for efficiency and lifetime frequently motivate these devices to be driven by pulsed cur-

rents. In the far-infrared, LEDs are again highly inefficient and sufficiently sensitive

to junction temperature to require external thermo-electric cooling [21].

In short, regardless of emission wavelength, the basic thermal physics of an LED

is the same:

" Imperfect wall-plug efficiency leads to self-heating that increases the device's

operating temperature.

" Elevated temperatures lead to decreased efficiency, regardless of whether self-

heating or ambient conditions are responsible for them.

On the other hand, if the light-emitting diode is examined as a thermodynamic

device, the exact opposite would be expected. Since the LED is driven by entropy-

20

free electrical power and results in the emission of entropy-carrying incoherent light,

it is possible for the device to absorb entropy from the ambient (i.e. self-cooling).

Moreover, since for a given spectral intensity of incoherent light output the outgoing

photon modes are occupied at some finite temperature, increased junction tempera-

tures should reduce the thermal gradient against which an LED must pump heat and

thereby permit higher efficiency.

The observed behavior of modern LEDs differs from these thermodynamic be-

haviors because even state-of-the-art emitters are far from their ideal limits. In this

work, we offer a theoretical framework to explain this discrepancy, present experi-

mental and numerical results to support it, and explore practical changes to device

designs to make LEDs more ideal.

1.2 The LED as a Thermodynamic Heat Engine

In Statistical Mechanics, the word "heat" is used to refer to any form of energy

which possesses entropy [22]. This usage applies equally to forms of energy referred

to colloquially as "heat," such as the kinetic energy in the relative motion of the

molecules in a gas or the constant vibrations of atoms in a crystal lattice, as well as

those for which the entropy is frequently less relevant, such as the kinetic energy in the

relative motion of electrons and holes in a semiconductor or the thermal vibrations

of the electromagnetic field in free space. Critically, the Laws of Thermodynamics

which govern the flow of heat are formulated independently of the Laws which govern

the deterministic trajectories of mechanical systems, be they classical or quantum.

As a result concepts such as the Carnot limits for the efficiency of various energy

conversion processes apply equally well to the gases and solid cylinder walls of an

internal combustion engine as to the electrons, holes, and photons in a modern LED.

An LED is an electronic device which takes entropy-free electrical work as input

and emits incoherent light which carries entropy. Instead of irreversibly generating

the entropy that it ejects into the photon reservoir, an LED may absorb it from

another reservoir at finite temperature, such as the phonon bath. As the diagram in

21

Figure 1-3 suggests, the device may absorb heat from the phonon bath and deposit

it into the photon field in much the same way as a Thermo-Electric Cooler (TEC)

absorbs heat from the cold side of the module and deposits it on the hot side [23, 24].

In the reversible limit the flows of energy and entropy are highly analogous for an LED

and a TEC. Moreover, in both the LED and TEC, the Peltier effect is responsible

for the absorption of heat from the reservoir being cooled into the electronic system

[25, 26, 27, 28]. Electrical work is being used to pump entropy from one reservoir to

another instead of simply creating it though irreversible processes. The LED, like the

TEC, is a thermodynamic machine.

For each bit of entropy JS absorbed on net from the phonon reservoir at finite

temperature, an amount of heat TatticeJS comes with it. Since input and output power

must balance in steady-state, the rate at which this heat and the input electrical work

enter the system (both measured in Watts) must exactly equal the rate at which heat

is ejected into the photon reservoir (also measured in Watts). That is to say, when

lattice heat is being absorbed an LED's wall-plug efficiency r7 (or equivalently, its

heating coefficient of performance), defined as the ratio of output optical power to

input electrical power, must exceed unity.

Additionally, in this picture the lattice remains slightly cooled compared to its

surroundings, so that heat is continuously conducted into the device from the environ-

ment in steady-state. Rather than self-heating, the LED is experiencing self-cooling.

The Second Law of Thermodynamics (i.e. non-deletion of entropy) places a clear

limit on the maximum efficiency of an LED in this framework. To understand this

limit, we must first understand the thermodynamics of photon gases at finite tem-

perature.

Incoherent electromagnetic radiation which originates in an LED is equally capable

of carrying entropy with it as electromagnetic radiation from a hot blackbody. All

incoherent light is therefore, in the statistical-mechanical sense described above, a

type of heat. The ratio of the rate at which radiation carries away energy to the rate

22

4TEC cold side

IrreversibleEntropy

Generation

Joule Heating& Thermal

Conduction

nativeRecobntion

Figure 1-3: Diagrams depicting energy and entropy flows in two types of thermody-namic heat pumps: TECs (top row) and LEDs (bottom row). The left column showsthe theoretical energy and entropy flows in Carnot-efficient devices. The right columnshows the same in devices with common sources of irreversibility.

23

Photon Field Phntnn FiPId

IrreversibleEntropy

Generation

Pnonon iela Phonon Field

I

at which it carries away entropy gives its flux temperature [29]:

dUldtTF - U(1-3)S dS/dt

Although this notion of temperature may be used to calculate the thermodynamic

limits of power-conversion efficiency, the rate of entropy flux in light is difficult to

measure directly. Fortunately a more intuitive definition of the temperature of light

is presented in Figure 1-4.

Consider two bodies that are each perfectly thermally isolated from their envi-

ronments (i.e. by adiabatic walls) and similarly isolated from each other. Suppose

body 1 has energy U1 and entropy S, and likewise the second body has energy U2

and entropy S2. If the insulating boundary between bodies 1 and 2 is replaced with

one which permits the flow of energy, the total energy U + U2 will flow to rearrange

itself in the way which maximizes the total entropy. The flow will stop only when

the addition of a differential amount of energy 6U to either body results in the same

fractional increase in the number of available micro-states for that body (i.e. the

same increase in its entropy). Equivalently, we may say that the flow of energy stops

when the bodies have equal temperature [22]:

aS1 1 1 _S 2

OU1 T1 T2 OU 2 (14

Now consider a similar scenario in which body 1 is an LED and body 2 is a perfect

blackbody radiator. To begin, both bodies are adiabatically isolated from their en-

vironments and each other. In the case of the LED, this means that the walls must

be perfect mirrors, such that each photon emitted eventually returns to generate a

quantum of reverse-current. Assume no non-radiative recombination occurs. The

LED is "on," but is in steady-state and consumes no power. Assume that the bodies

have no means of exchanging energy other than through photons and that to begin

the boundary between them is also a perfectly-reflective mirror.

If the mirror is modified to permit transmission over a narrow range of wavelengths

24

Body I Body 2

Adiabatic Boundar(no heat exchnge)]

Body 1 Body 2

Bodies re-arrange energy U1+U, to maximize total entropy Si+S,-Equilibrium reached when:

T IS 2 1I dU 9U2 2

Body 1 Body 2

Bodies re-arrange energy U1+U2 to maximize total enrropy !S+S2.Equilibrium reached when:

-1 j dS -1SaU = a2du, adl 2

Body 1: LED Body 2: Hot blackbody

Perfect MirrjAdiabatic Boundary

(no heat exchange)

Body 1: LED Body 2: Hot Blackbody

A-Selective Mirror

100% 1i1 - - Transmission %

Reflection %

Equilibrium: Zero Net Photon Flux%1t Mrahiw

0 n

Figure 1-4: The brightness temperature of an incoherent source (here, an LED) may

be defined as follows. At each optical frequency, consider the temperature at which

a perfect blackbody would emit with the same spectral intensity (i.e. power per unit

area per unit frequency). This temperature indicates the ratio of the rate of energy

flux to the rate of entropy flux carried by the radiation in that band. The weighted

average of these temperatures over the intensity spectrum of the emitter gives the

brightness-temperature of the source.

25

around A0 , energy will flow on net from the body with higher spectral power density

normal to the boundary (i.e. I(A) in W m- 2 nm- 1 ) to the body with lower I(A)

at A0 . If we assume the LED is perfectly incoherent, the flow of photons in either

direction is equally capable of carrying entropy, and therefore equally justified in

being termed 'heat.' Since heat may only flow from high temperature to low, the

equilibrium condition for the two bodies may only be satisfied when 11 (Ao) = 12 (Ao).

Since the relationship between intensity and temperature for a perfect blackbody is

given by the Planck radiation law, we may define the brightness temperature TB of

any completely incoherent source as the temperature of blackbody whose spectral

intensity equals that of the emitter in the wavelength range of interest [29, 30]:

4h7r2 C2 IIemitter(AO) = Iblackbody (AO ;TB) h(27rc/Ao) (1.5)

0 exp (h(kBT/A) - 1kBTB/

Note that unlike the color temperature of radiation commonly used in the light-

ing and display industries, a longer-wavelength emitter is not necessarily cooler than

a short-wavelength emitter. Both the linewidth and intensity of the source matter and

may result in thermodynamically-cold emission from a blue LED or thermodynamically-

hot emission from a red one. The flux temperature TF and brightness temperature

TB of a source may be cool, even when the radiation is blue.

A note to the reader: a more detailed discussion of the distinction between the

flux (TF) and brightness (TB) photon temperatures can be found in Appendix A.

Since the temperature of an incoherent photon flux is essentially a measure of

its spectral intensity I(A), the Second Law places a different efficiency constraint

on emitters of different spectral intensity. As a function of lattice temperature and

emitter intensity, the Carnot limit may be expressed compactly as follows:

77 77Carnot = Tphoton (I)Tphoton (I) - Tattice 1.6

For bright sources (I(A) > Iblackbody(A; Tattice)), the LED must pump heat against

the large temperature difference between the lattice and the outgoing photon field.

26

This results in a maximum efficiency, even for a perfect Carnot-efficient LED, which

exceeds unity but only slightly. For dim sources (I(A) - Iblackbody(A; Tiattice) <

Iblackbody (A; Tattice)), the LED must only pump heat against a small temperature

difference. As a result, efficiencies far in excess of unity are possible.

Examination of Equation 1.6 at fixed spectral intensity I reveals another counter-

intuitive aspect of the heat-pump regime. As Tattice is increased, the temperature

difference against which the LED must pump becomes smaller, and the maximum

allowable efficiency increases.

Thus, the basic thermal physics of an LED in the heat pump regime is the reverse

of the conventional thermal physics:

" Above-unity efficiency results in self-cooling that decreases the device's operat-

ing temperature.

" For a desired spectral intensity, a higher lattice temperature means that the

device can be more efficient.

These differences may result in practical consequences for both the device-level

design of LED active regions (explored in § 2.5) and the thermal design of their

packaging (which we discuss briefly in Chapter 6).

1.3 Previous Work Toward Unity Efficiency

For several decades it has been theoretically understood that the presence of entropy

in incoherent electromagnetic radiation theoretically permits semiconductor light-

emitting diodes (LEDs) to emit more optical power than they consume in electrical

power [31, 29, 32, 33]. Moreover, starting very early on the phenomenon has drawn

the attention of the applied research community. In 1959 a US Patent was granted

for a refrigeration device based on the principle [34]. In the last decade, the applied

literature on the subject has expanded to include more realistic modeling and more

recent advances in device fabrication technologies [14, 35, 36, 37, 38, 39] and at least

one attempt to demonstrate practical cooling is currently underway [40]. Nevertheless,

27

prior to this work, the basic phenomenon of electrically-driven light emission above

unity efficiency had never been experimentally verified.

The experimental literature on electro-luminescent cooling stretches back more

than five decades, beginning before even the early work of Tauc [31] in 1957 and

Weinstein [291 in 1960. A summary of this work appears in Table 1.1 alongside data

for experiments described in this thesis.

Year Author(s) Vmin qVmin/kBT e~z/kBT Max Reported q

1953 Lehovec, et al. [41] 1.8 V 70 < 2.5 x 10-34 Not Published

1964 Dousmanis, et al. [42] 1.25 V 186 2.8 x 10-90 16% [43]

1966 Nathan, et al. [44] 1.1 V 6380 10-36o 6 %

2005 Wang, et al. [4.5] 0.36 V 14.2 3.8 x 10- Not Published

2011 Oksanen, et al. [40, 46] 0.5 V* 19.3* 4 x 10-13 Not Published

2011 THIS WORK (§ 3.2, [47]) 70 uV 0.002 8.4 x 10- 7 231 ± 37 %

Table 1.1: Summary of previous experiments towards electro-luminescent cooling

(i.e. electro-luminescence with q >1). The asterisk (*) indicates that these figures

were taken from simulation data. The quantity qVmin/kBT highlights the primary

difference between the approach taken in this work and previous experiments. The

quantity e-h/kBT provides a scale for the optical power available in the low-bias

regime.

As early as 1953, Lehovec et al. speculated on the role of thermo-electric heat

exchange in SiC LEDs [41]. The authors were motivated by their observation of

light emission with photon energy hw on the order of the electrical input energy per

electron, given by the product of the electron's charge q and the bias voltage V.

In 1964, Dousmanis et al. demonstrated that a GaAs diode could produce electro-

luminescence with an average photon energy 3% greater than qV [42]. Still, net

cooling was not achieved due to non-radiative recombination [43] and the authors

concluded that the fraction of current resulting in escaping photons, typically called

the external quantum efficiency 17EQE, must be large to observe net cooling. They

wrote:

28

"Diodes with high quantum yield are required for direct experimental

observation of the cooling effect."

DOUSMANIS, ET. AL.

PHYSICAL REVIEW, 1964

A similar observation was made two years later in a cryogenic GaAs LED (hw =1.44eV)

by Nathan, et al [44]. Then after several decades of minimal experimental activity,

recent modeling and design efforts have indicated that EQE could be raised toward

unity by maximizing the fraction of recombination that is radiative [14, 35, 38] and

employing photon recycling to improve photon extraction [14, 35, 37]. As a result,

at least one experiment was performed by Wang, et al. in 2005 [45], but no opti-

cal power or wall-plug efficiency data was published. At least one effort to observe

electro-luminescent cooling with JEQE above 50% continues to be active [40], although

early results suggest problems with shunts in the emitting diode [46].

All of these experiments followed the logic of the quote above from Dousmanis,

et al. by attempting to raise %EQE toward unity. In contrast, q > 1 was observed in

this work with nEQE ~ 3 x 10- 4 . Since the wall-plug efficiency q of a diode may be

expressed as follows:

S=--EQE ,(1.7)qV

in order to achieve above-unity q with small 77EQE requires V < hw/q. Multiple

authors have dismissed such operating regimes in the past because of the low output

power available in this regime, but the present work has found it's consideration

worthwhile for 3 main reasons:

" Regardless of the power requirements for a practical cooling system, lower power

may be sufficient for specific applications and/or experimental confirmation.

" The greatest deviations from conventional 7 < 1 operation (i.e. highest coeffi-

cients of performance) always occur at low power. This is a general property of

endo-reversible heat pumps.

" The decrease in power from lowering V can be compensated by increasing the

29

ratio kBT hw.

The third observation above was made in 1985, when Paul Berdahl presented an

analysis of semiconductor diodes as radiant heat engines [43]. In that work, he showed

that the available cooling power decreased exponentially with the ratio of the diode

materials bandgap Egap to the thermal energy kBT, in accordance with the blackbody

emission power integrated over the absorptive/emissive band.

1.4 Efficient Communication with a Photonic Heat

Pump

The experimental result presented in Chapter 3 not only realizes photon generation

with wall-plug efficiency in excess of unity (i.e. net cooling), but further demonstrates

that arbitrarily high wall-plug efficiency is available at infinitesimal power. Data for

the generation of 2.47pm photons (w ~ 500 meV) in a 423 K environment (kBT

36 meV) appears in Figure 1-5.

At the low-power, high-efficiency endpoint of this data set, the LED consumes

just 8.8 meV of work per photon to create an optical signal which may be directly

electrically modulated. In principle, such a device could be used as the source in a

simple on-off-keying (OOK) communication link. If the emission of one such photon

were used to indicate a '1' and the lack thereof to indicate a '0', on average just 4.4

meV of work would be required per bit transmitted. This figure is well below the

accepted limit [48, 49] for efficient electromagnetic communication of kBT ln(2) per

bit (about 25 meV/bit at 423K).

This simple communication architecture ignores the substantial increase in bit-

error-rate (BER) that such a scheme would suffer due to thermal noise (i.e. blackbody

radiation), even with perfect collimation and a perfect receiver node. Unsurprisingly,

the kBT ln(2) limit for all electromagnetic systems is fundamentally connected to this

thermal noise; the limit and the power density of this noise source both vanish as T -+

0. In Chapter 4, we explore theoretically the limits of energy-efficient communication

30

with a Carnot-efficient heat pump in the presence of noise due to blackbody radiation.

Before that, however, it is constructive to review a few basic results from the extensive

literature on this topic.

Photoneryk...

C0

0 0.101~~0

kBT-In(2)

CLI 0.01

CL

0.00110 10 10

Photon Emission Rate (ifs)

Figure 1-5: At low power, a conventional LED may generate a photon with an arbi-trarily small amount of work. As with any endo-reversible heat pump, the efficiencyscales inversely with the output power resulting in the trade-off between photon emis-sion rate and per-photon work consumption. For low photon emission rates, theper-photon work has been experimentally observed to fall below kBT - log(2), raisinginteresting questions about the limits of efficient communication.

In 1948, Claude Shannon published a paper in the Bell System Technical Journal

entitled A Mathematical Theory of Communication [50]. The manuscript is often said

to have laid the conceptual groundwork for the digital revolution by proving that all

forms of digital and analog information could essentially be measured in the same

units- typically bits. In this same paper, Shannon proved that for a known physical

channel with known noise properties, one could calculate a maximum capacity for the

transmission of information per unit bandwidth.

In his paper, Shannon considered the problem of communication in the presence

of Additive White Gaussian Noise (AWGN). Interestingly, this noise distribution

corresponds to the thermal noise distribution for field variables (i.e. voltage V or

electric field E) in the quantum degenerate limit hw < kBT where most electronic

31

circuits and radio-frequency links operate [51]. For this type of noise source, the

following formula for channel capacity may be proven [50]:

C = Af log 2 ( + (1.8)

where C is the channel capacity in bits per second, Af is the bandwidth of the

channel, P is the average power of the signal, and N is the average noise power per

unit frequency within the channel's bandwidth. For a given noise power density (per

unit frequency), the formula indicates how much power must be present in the signal

field to communicate at a given rate C.

This result is typically associated with discussions of the fundamental energy re-

quirements for any physical process of communication. To see why, consider the

question of linear electro-magnetic communication using a single channel (i.e. a sin-

gle transverse mode with a single polarization state) in the presence of blackbody

radiation.

Assume the noise power N comes from thermal fluctuations of the electromagnetic

field and the frequencies of interest are assumed to be in the quantum degenerate

limit. Since the quanta become irrelevant in this limit, the field may be described

by classical statistical fluctuations so that for each mode, the average energy of the

fluctuating field is kBT by equipartition. Then if we consider a channel of length

L > c/f, the density of forward-traveling modes is simply 1/(2ir/L) in k-space or

L/c in f. Combining this information, we arrive at the thermal energy density per

unit frequency in the channel of length L:

U L-- = - kBT (1.9)

Af C

Since this channel empties its thermal energy at the receiver end in time At = L/c,

the noise power is simply:

U L cN =A-= -kBT - = kBT. 1.10

Af c L

32

Substituting this expression into the channel capacity formula above allows us to

relate the rate of information flow C to the rate of energy flow in our signal P:

C = Af log 2 1 + k ). (1.11)kBTA f

The maximum ratio of C to P appears at low power, where the logarithm can be

expanded to give the minimum energy per bit transmitted under these assumptions:

min (- = kBT ln(2). (1.12)

It has been pointed out by numerous authors [49, 52, 53] that this formula does

not imply that there is a minimum energy cost to communication. The canonical

example is mailing a hard drive. Considering this example recasts communication as

a choice of reference frame rather than a physical process. In contrast, many authors

have come to the conclusion that the operation of erasure does appear to carry with

it an unavoidable energy cost [49].

Over the years, several authors have used specific examples to point out the rela-

tionship between the kBT hn(2) result and the assumptions that went into its deriva-

tion. One commonly pointed out assumption is that of the field's linearity with

respect to the addition of noise to the signal [49, 52].

In this work, we point out another assumption which we believe may not have

been previously raised. We point out that there is a distinction between the rate of

flow of entropy-free work into a source and the outflow of electromagnetic energy.

One immediate question of interest presents itself: is there a limit to the ratio of an

emitter's work expenditure rate to the information flow rate it may encode:

min - m = mm - ? (1.13)C bit

For the emitter involved in the electro-luminescent cooling experiment, operation

below the cooling power peak allows the ratio of power to work consumption rate P/W

to be arbitrarily large. As a result, it presents a surprisingly accessible platform for

33

Classic AWGN Symbol Space Heat Pump Symbol Space--1 Bit-1B

0.1 -0 BIt 0.1

CD 0co 00 0

-0.1 -0.1

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1Time (s) Time (s)

I B# -1 Bit

04 04

-82 -0.1 0 0.1 0.2 ..8.2 -0. 0 . 0.2Voltage (V) Voitage (V)

Figure 1-6: Typical members of the symbol spacedormee nunication in the presenceof thermal electromagnetic noise. The left column shows two representations of a pairof symbols for communication with a conventional signal. The right column showstwo representations of a pair of symbols for communication with a heat pump.

experimentally exploring the limits of energy-efficient electromagnetic communica-

tion in the non-degenerate noise hw > kBT limit. Naive interpretation of this fact

combined with Equation 1.8 suggests that arbitrarily efficient communication should

be possible using a heat pump.

Upon closer examination, however, the signal generated by a heat pump may

be arbitrarily efficient in the power-conversion sense (i.e. many symbols per unit

energy), but the '1' symbol produced in the efficient regime is less distinguishable

from the '0' symbol and therefore leads to less information flow (i.e. fewer bits

communicated per symbol transmitted). This is because the '1' symbols it transmits

are composed of a different distortion of the photon field from thermal equilibrium,

as shown in Figure 1-6. Interestingly, this result suggests a fundamental trade-off

between the disorder required for efficient heat-pumping and the distinguishability of

the symbols in the codebook, leading to a direct connection between the information-

theoretic entropy of a source and the physical entropy exiting the apparatus used

to communicate it. A thorough information-theoretic analysis of this trade-off is

34

presented in Chapter 4.

1.5 Potential Practical Applications

The basic result of an LED operating as a heat pump also holds consequences for

several potential practical applications.

1.5.1 Infrared Spectroscopy

As seen in Figure 1-7, several common molecules have distinct absorption features in

the mid-infrared wavelength range. For this reason, substantial attention has been

given to developing sources for absorption spectroscopy here [54, 12, 5, 35, 10, 56].

I.00

wavenumber (cmn )

2223

NOMethan

CO 2

NN2

wavelength (pfr)

Figure 1-7: Numerous abundant molecules have absorption features in the mid-infrared wavelength range, making it a valuable band for spectroscopy. This figure istaken directly from Figure la of Reference [54].

A review of the available emitters appears in Table 1.2. Two main types of emit-

ters are available: thermal emitters and light-emitting diodes. Thermal emitters are

efficient, but emit over a wide wavelength range and carry long thermal time con-

stants which limit their direct switching speeds. Mid-IR LEDs may be switched at

35

Type Producer Model No. Wall-Plug Emitted Modulation

Efficiency Power Frequency

Thermal HelioWorks EP3872 0.15 % 3.5 mW 2 Hz

Thermal HawkEye Tech- IR-55 R 0.29 % 2.7 mW 10 Hznology

Thermal IonOptics (ICX NL8LNC 0.25 % 5.6 mW 5 HzPhotonics)

Thermal IonOptics (ICX Tun IR 0.20 % 0.27 mW 1 HzPhotonics)

Thermal Heimann Sensor HSL EMIR- 0.31 % 1.4 mW 10 Hz2000R

Thermal Intex MTRL-17- 0.39 % 3.8 mW 15 Hz900R

LED ICO Ltd LED-42 0.01 % 0.01 mW 100 kHz(RMT)www.optico.ru

LED IoffeLED LED42Sc 0.15 % 0.03 mW 10 MHz

LED Roithner LED-43 0.013 % 0.01 mW 10 MHz

Table 1.2: Comparison of existing sources for spectroscopy around A = 4.25pm. Note

that wall-plug efficiency and emitted power consider only the power within a spectral

band of AA = 0.45pm and an angular cone of 300. Table adapted from [4].

36

hundreds of MHz, allowing the use of lock-in techniques to improve the Limit-of-

Detection (LOD) [57, 55], but are relatively inefficient in terms of power conversion.

However, since LED emission is concentrated at photon energies just above the ma-

terial bandgap, the so-called "spectral wall-plug efficiency" (i.e. wall-plug efficiency

considering only emitted photons in a narrow band of spectroscopic interest) of an

LED can be competitive with that of a thermal source. Analysis of the characteristics

required for spectroscopy suggests that a reasonable figure-of-merit for an opto-pair

(emitter-detector pair) system is the so-called "normalized LOD," measured in parts

per million per mW of source drive power per Is of lock-in time constant [4], and

suggests that LED-based spectroscopy systems are substantially superior to those

that use thermal emitters.

High-Temperature Environments

Although infrared LEDs can be designed to emit at a variety of wavelengths of spec-

troscopic interest [54, 21] and may be directly modulated at the high frequencies

employed by lock-in techniques [4], conventional devices suffer from carrier leakage

and Auger recombination which limit their utility at high temperatures. Unfortu-

nately many of the largest applications for such spectroscopy tools are tied to such

harsh environments.

For example, radiation near A =3.3ptm is strongly absorbed by methane, so LEDs

at this wavelength could be used for downhole oilfield spectroscopy. As the pace of

discovery of new oilfields has diminished, oil companies have been forced to focus on

upgrading existing ones to meet rising global demand. To accomplish this efficiently,

they must avoid costly errors in the design of surface extraction facilities and capital

misallocation caused by inaccurate reserve estimates. As a result, renewed focus has

fallen on developing platforms for in-situ determination of the gas-to-oil ratio (GOR)

in the hydrocarbon-rich fluids present downhole in an oilfield [58]. However, spectral

data in the mid-infrared has been unavailable downhole due to a lack of sources and

detectors for this purpose [59]. An efficient, fast-switching source at 3.3pm could

benefit such a spectroscopy system if it were capable of operating at temperatures of

37

175-200'C and pressures >100 MPa.

Radiation near A =4.2[pm and A =4.7pim are strongly absorbed by carbon dioxide

and carbon monoxide respectively. Spectroscopic analysis in these bands could be

used to determine the composition of combustion products. The extreme temper-

atures found in vehicle exhaust and industrial flue gases (as well as the machinery

around them) may require high-temperature performance for sources intended to per-

form these operations in situ [601.

Ultra-Low-Power Systems

Recent advances in the efficiency of micro-electronic circuits have enabled a new

generation of ultra-low-power sensor and display systems based on LEDs. Here,

the LEDs are frequently the primary load, and therefore constrain the mobility and

lifetime of the overall systems.

For example, the power budget of an ultra-low-power pulse oximetry system devel-

oped in 2010 appears in Figure 1-8 [3]. Here, the differential absorption of two LEDs

(one at 660nm and another at 940) is used to detect the oxygen concentration of the

blood in a patient's finger. Over 90% of the total power in this system is devoted

to the LEDs and their associated switching control circuits. The authors consider

this to be practically valuable because it permits a single set of 4 AAA batteries

to operate the sensor for up to 60 days. While this is more than 10x longer than

other implementations, if the 660nm and 940nm photons could be generated twice as

efficiently, the operating lifetime between charges could be increased to 120 days.

The requirements for the brightness and wavelength of the source in this pulse-

oximetry system are significantly less demanding than in general-purpose indoor il-

lumination. It therefore seems likely that any improvements to state-of-the-art LED

efficiency resulting from design for low-voltage operation will benefit ultra-low-power

systems before they are relevant for solid-state lighting.

38

Less than 4001W of

processing power

Figure 1-8: Recent advances in efficient amplification circuitry leave state-of-the-artultra-low-power pulse oximetry systems with power budgets dominated by inefficientred and infrared LEDs. Figure taken from Table II of Reference [3].

1.5.2 Lighting

Recent advances in solid-state lighting have made available LEDs capable of convert-

ing electrical power into white visible optical power above 50% wall-plug efficiency

[61, 62] with further improvement anticipated in the coming years [63]. These results,

however, are typically achieved with pulsed operation, where the emitting diode does

not heat up. So-called "hot" steady-state testing leads to substantially diminished

efficiency [7]. As discussed in § 1.1, the ubiquitous loss mechanisms of non-radiative

Auger recombination and carrier leakage are largely responsible [18, 19].

Experimental confirmation of electrical-to-optical power-conversion efficiency in

excess of unity raises the possibility of building electrical light bulbs with no net

waste heat generation [37]. Not only would such bulbs be highly efficient, but they

could result in large cost savings from the removal of heat sinks that dominate material

costs and improvements in bulb lifetime due to the abatement of thermally-accelerated

failures of driver components such as electrolytic capacitors [73.

Although the work in this thesis focuses on devices which emit outside the visible

39

Power Consumption per Block Value

Oscillator/LED & Switching Control 4.4mW

Two Transimpedance Amplifiers 80pW

Two Low-pass Fitters 300AW

Ratio Computation 2.2pW

Rferene Otnerator/Bias Circuitry 1 1.5pWTotal 4.8mW

spectrum, we discuss the future of solid-state lighting technology in light of our results

in § 6.4.

1.5.3 Other Applications

The net absorption of heat from the emitter's lattice combined with the ease of

achieving long ballistic path lengths for infrared photons in semiconductors makes

electro-luminescence interesting as a solid-state cooling technology [36, 37, 38, 39, 40,

34].

Less widely-discussed but conceptually related is a generalization of Thermo-

Photo-Voltaic (TPV) electrical power generation known as Thermophotonics (TPX)

[64, 65, 66]. Here, the passive narrow-band-emitting surface of the TPV is replaced

with an active device, an LED. When V = 0, the passive and active emitters have

identical performance, but as a forward bias is applied, the emitted power rises more

rapidly than the input power. In fact, since this ratio diverges as V -+ 0 and the

efficiency of extracting work from those photons is nonzero (Temitter > Tabsorber), the

maximum net output power (i.e. electrical power from the photo-voltaic minus LED

drive power) is guaranteed to take place at V > 0. Nevertheless, emitter surface ma-

terials are chosen based on other criteria, for example their high-temperature stability

and ease of patterning into photonic crystals, and new constraints would be placed

on them by the need to make the emitting surface a direct-gap semiconductor inside

a diode structure. In light of these constraints, it is likely that high %QE emitters

would be required to improve TPV performance.

Finally, LEDs with extremely high wall-plug efficiency may be useful for free-space

communication by satellites. When these power-constrained satellites send signals to

the ground, the power consumed in reconstructing the signal is much cheaper than

the power consumed in transmitting it. As a result, the constraints on these systems

closely approximate the problem of encoding information into the outgoing electro-

magnetic field with a minimum of electrical power consumption. In fact, schemes

such as Pulse-Position Modulation (PPM), which allows multiple bits of informa-

tion to be communicated per photon [67], find application here. Moreover, the long

40

wavelengths of certain "atmospheric windows" [56] for which efficient lasers are not

available correspond to wavelengths at which heat pumping LEDs theoretically emit

more power at high efficiency. As a result, the LED technologies developed in this

work may prove useful for such niche communication systems.

1.6 Thesis Outline

In Chapter 2 we use various simplified device models to explain LED operation above

unity efficiency and explore device design concepts intended to push LED perfor-

mance toward the limits imposed by The Second Law. In Chapter 3 we validate

aspects of this framework through a series of experiments on existing devices. In

Chapter 4, we explore the ultimate consequences of these design improvements for

photonic communication by exploring the physical limits of energy-efficient commu-

nication with a heat pump. In Chapter 5, we consider the practical applications

of these thermo-electrically pumped LEDs to power-constrained infrared absorption

spectroscopy systems operating in high-temperature environments.

41


42

Chapter 2

LEDs as Heat Pumps

In this chapter we explore the thermodynamic behavior of LEDs and construct a

framework for their analysis as heat pumps. In § 2.1, we review electron transport

in an LED with emphasis on the flow of entropy. Then in § 2.2, we organize these

flows within a basic model of an LED as a thermodynamic heat pump. In § 2.3,

we explain why all LEDs should in theory act as heat pumps at low forward bias

voltage. In § 2.4 we analyze an idealized reversible LED and discuss its relationship

to existing devices. We find that an ideal LED achieves the Carnot efficiency and that

both ideal and non-ideal LEDs face the same trade-off between power and efficiency

that all thermodynamic machines operating at nonzero power experience. Then in

§ 2.5 we present initial work on the design of LEDs for efficient operation in the heat

pumping regime. Finally, in § 2.6, we generalize this framework to describe the flow

of electrons around a closed circuit as the flow of a working fluid through a closed

thermodynamic cycle.

2.1 Electron 'ransport and Entropy Flow in LEDs

Although the claim of steady-state electrical-to-optical energy conversion at above-

unity efficiency may appear to violate the Laws of Thermodynamics, it is not only

consistent with them, it's presence at low power is a fundamental property of any

LED. The issue of energy conservation in q > 1 operation (i.e. the First Law issue)

43

is resolved by the inclusion of lattice heat absorption within the diode.

This explanation immediately raises a question about consistency with the Second

Law of Thermodynamics. Because the vibrational energy of the lattice is heat, the

net absorption of energy from the lattice must be associated with a net absorption

of entropy as well. The issue of entropy non-deletion (i.e. the Second Law issue) is

resolved by the entropy associated with the emitted incoherent photons. That is to

say, for a bounding surface drawn around an LED operating at r7 > 1, the net inflow

of entropy due to lattice heat absorption is offset by an outflow of at least as much

entropy through the photons. Furthermore we may calculate the inflow and outflow

of entropy to the electron-hole subsystem due to thermally-assisted carrier injection

and radiative recombination, and thereby provide a more mechanistic explanation of

how the device transports the absorbed entropy from the lattice to the photon field.

By calculating these entropy flows we arrive at a more complete model of device

operation which complies with a continuity equation for entropy flux. That is to say,

we may show that our model of LED operation is not only globally consistent with

The Second Law, it is locally consistent as well.

2.1.1 Current Continuity

As depicted in Figure 2-1, a conventional double hetero-junction light-emitting diode

consists of a layer of narrow-bandgap intrinsic semiconductor sandwiched between a

pair of wider-gap layers. The wider-gap layers are doped p and n-type and have metal

contacts attached to form the positive and negative electrical terminals of the device

respectively. When a forward voltage is applied, electrons from the n side and holes

from the p side are injected into the active region. There they undergo recombination

through various mechanisms, connecting the electron-type current from the n side

with hole-type current on the p side to satisfy current continuity. Although some

leakage does occur (i.e. minority carriers diffusing across the double-line boundary

in Figure 2-1), the hetero-structures are designed to minimize this. Since the basic

transport processes can still be understood while neglecting leakage, we will do so

here in § 2.1. In our simplified picture then, for each quantum of charge that flows

44

LUE) E*

Position x

Figure 2-1: Simple band diagram for a double hetero-junction LED at low forwardbias. The basic transport processes described in § 2.1 are overlaid for the reader'sconvenience. The double-line with diamonds is a fictitious boundary that we assume

is crossed only by recombining carriers in this simplified analysis.

45

between the terminals of a diode at sub-bandgap forward bias voltage, the following

three processes must take place:

" One electron must escape the cathode, traverse the n-doped quasi-neutral re-

gion, enter the intrinsic active region, and climb a potential energy barrier to

the recombination site.

" One hole must escape the anode, traverse the p-doped quasi-neutral region,

enter the intrinsic active region, and climb a potential energy barrier to the

recombination site.

" The electron and the hole must recombine by some process which conserves

energy and momentum.

The first two processes are referred to as the thermally-assisted injection of elec-

trons and holes respectively. The last is recombination. After a short introduction to

the concept of quasi-equilibrium, we will proceed to analyze these two processes to

complete our picture of electron transport in this simplified model.

2.1.2 Quasi-Equilibrium

Electronic transport in these devices is typically described in the framework of quasi-

equilibrium. In quasi-equilibrium, the single-particle states in a given band and region

of the device are taken to be in sufficiently close contact to be occupied according to

some Fermi-Dirac distribution, with some Fermi level EF and some temperature T.

Typically this assumption is justified by the fast phonon scattering present in common

semiconductors at room temperature. Typical timescales for carrier momentum and

energy relaxation are on the order of picoseconds and nanoseconds respectively, while

the timescales for carrier diffusion processes connecting different regions of devices

with micron-scale features are much slower [68]. Thus, under the assumption of quasi-

equilibrium, specification of EF,e(X), Te(x), EF,h(X), Th(x) at each point constitutes

a complete description of transport within a device.

46

Moreover, fast phonon scattering typically limits differences between the carrier

temperatures and the temperature of lattice. As a result, it is common to see band

diagrams which depict only EF,e(v) and EF,h(x) across the device, and assume

Te (X) = Th (x) = Tlattice - (2.1)

The concept of quasi-equilibrium is useful in great part because (in isothermal

systems), carriers only flow in response to differences in EF. When two adjacent

points in space have different electrical potential, an electric field is present a drift

flux of carriers occurs in response to it. Likewise, when two adjacent points in space

have a different number density of carriers, diffusion leads to a flux from high density

to low. The quasi-Fermi level combines these two processes in such a way that a

flat EF,e(X) indicates that the electron drift and diffusion fluxes are balanced and

offsetting. That is to say, the conduction states at the points in space where EFe(X)

is flat may be considered to be in equilibrium. And of course the same is true for

holes when EF,h(x) is flat.

On the other hand, when EF,e(x) and EF,h(x) are not flat, adjacent positions are

not in equilibrium. Gradients of quasi-Fermi levels within the conduction band drive

electron flows, and likewise for the valence band and hole flows. A difference between

the Fermi levels for the two bands at the same point in space drives generation or

recombination.

VEF,e(x) # 0 -> Electron Flux (2.2)

VEF,h(x) 4 0 - Hole Flux (2.3)

EF,e - EF,h > 0 -- Recombination (2.4)

2.1.3 Thermally-Assisted Injection

With no applied voltage, all Fermi levels at all positions remain equal. There is no

net injection to the active region. Still, the gas of electrons on the n side (and holes

on the p side) is perpetually emitting and absorbing phonons to exchange energy and

47

entropy with the lattice vibrational modes. These processes are in equilibrium when

the amount of entropy added by a small addition of energy to each system is equal

(i.e. they are at the same temperature).

When a small forward bias voltage is applied, the potential energy of electrons at

the n-contact becomes higher than those at the p contact. As a result, VEF,e(x) and

VEF,h(x) become nonzero and net flows of these carriers occur as shown in Figure 2-1.

In order for an electron to flow from the states relevant for conduction at the

n-contact (x = L) to those relevant at a recombination site (x = XR), it must climb

a potential barrier. The same is true for a hole from the p-contact (x = 0). As is

readily seen from the figure, the combined heights of these two barriers, AVeiectrons

and AVholes, are simply related to the natural energy scales of the problem.

AVeectrons + AVholes = Egap,active - qV + O(kBT) , where (2.5)XR

AVholes = Hmetal,p + ]R -VEv(x)dx + O(kBT) and (2.6)

AVeectrons = j VEc(x)dx + Rn,metal + O(kBT) (2.7)

Here Ec (x) and Ev (x) denote the conduction and valence band energies respectively,

and r1 a,b denotes the Peltier coefficient at the metal-semiconductor interface with

material a at left and b at right. The terms of order kBT are present because elec-

tron transport occurs within the conduction band rather than at the band-edge, and

likewise for holes. Because kBT < Egap,active, they will not figure prominently in our

analysis here.

So where does an electron get the energy to climb this barrier? The answer is

that lattice heat is absorbed all along the x = 0 to x = xR path by means of the

Peltier effect. Typically the Peltier effect is described as thermo-electric effect at an

interface: when an electric current I crosses from some solid material a to another

material b, heat is removed from the lattice vibrations in the vicinity of the interface

at a rate Q proportional to the current:

Q = A,BI . (2.8)

48

Re-thermalization

4 G e flux

D G

u.- -E-

F~eLU

-20 0.5 1

Figure 2-2: The Peltier effect at an interface. Hot carriers are thermionically emittedover a hetero-junction barrier. A re-thermalization process ensures that the electronsin material a remain in a thermal distribution. This process requires the absorptionof approximately AEc of lattice thermal energy per electron, resulting in so-called"Peltier cooling."

A more physical picture of the Peltier effect is found in Figure 2-2, and may be

readily generalized to conduction away from interfaces. In quasi-equilibrium, conduc-

tion between two points in a given band of a given solid can be ascribed to a difference

in EF between those points. Consider Figure 2-3a. If we discretize space and consider

adjacent points, we see that in a region with an electric field (i.e. VEc # 0), there is

both a finite difference in the Fermi energy as well as a finite difference in the energy

of the conduction states available for transport. Using this procedure, we can see

that the transport in Figure 2-3b will also lead to lattice heat absorption.

Generally speaking, whenever the direction of carrier flow f/q opposes the electric

force qE, the Peltier effect causes the carrier population to absorb heat from the

lattice. This is exactly the situation in Figure 2-1. The amount of heat absorbed

across the device during the thermally-assisted injection of each electron-hole pair is

49

Re-thermalization Re-thermalization

4f ( eflul

Ece EC

---------- Ee -_--.... ~

(a) Discrete model. (b) Continuous model.

Figure 2-3: Models for electron transport in a region where the Fermi level gradientdrives a net carrier flow against the direction of electric field drift. The continuous-space model at right is a generalized version of the Peltier effect shown in Figure 2-2.The generalization is intuitive when the quasi-equilibrium concept is applied, as isdone in the model at left.

equal to the height of the potential barrier they must climb. In fact, a Peltier term

I corresponding to the thermal energy absorbed per pair may be substituted into

Equation 2.5 to give:

Egap,active - qV AVeectrons + AVhoes (2.9)

2.1.4 Recombination

The final transport process required to maintain current continuity is the recombi-

nation of injected electrons and holes. Although some leakage happens in any real

device, for simplicity we consider only recombination sites in the active region.

As with the majority carriers in the doped regions, even when the device is off the

electrons and holes in the active region are perpetually experiencing generation and

recombination as the result of their interaction with other reservoirs. These processes

can be thought of in terms of the following chemical reaction equation:

e~ + h+ ( ) Ubandgap (2.10)

50

where e- is an electron, h+ is a hole, and Ubandgap denotes some excitation with

energy (and other conserved quantities) equal to that of the electron-hole pair. As

with any chemical reaction, the reactants and products are in equilibrium at some

concentrations. When the concentration of electrons n and holes p exceeds these

values (i.e. when np exceeds the squared intrinsic carrier concentration n?), net

recombination occurs and the reaction in Equation 2.10 is driven from left to right.

Likewise, when n and p are below their equilibrium values, net generation occurs and

the reaction is driven backwards.

Each time that an electron-hole pair is annihilated, both energy and entropy are

removed from the electron and hole gases. That is to say, the number of microscopic

configurations in which the conduction and valence bands can be occupied is reduced.

This entropy, however, cannot disappear entirely. Doing so would violate The Second

Law. Instead, the entropy which is removed from the electronic sub-system (i.e.

the degrees of freedom from excitations of the conduction and valence band states)

is transported to another sub-system at the same location in the device. Which

sub-system that is depends on where the electron-hole pair's energy went. For non-

radiative recombination, the destination is the lattice. For radiative recombination,

the destination is the photon field.

When a non-radiative recombination event occurs, Ubandgap is deposited into the

lattice excitation spectrum. These new excitations allow the lattice to inhabit a larger

space of microscopic configurations and thereby increase the entropy of the phonon

field. The amount of entropy (AS)iattice may be calculated simply by making use of

the lattice temperature Tattice:

(AS)Iattice = Ubandgap (.1Tiattice

The same is true for the photon field. The number of possible microscopic configu-

rations of the photon field also increases. In fact, the definition of brightness tem-

perature given in Equation 1.5 was designed specifically to quantify the additional

51

entropy (AS)phtOnl that appears in the photon field with Ubandgap:

(AS)photon Ubangap (2.12)Tphoton

2.1.5 Continuity of Entropy Flux

When an LED is put into a forward bias condition such as shown in Figure 2-1, it is

taken out of equilibrium. After a short time, its sub-systems approach a condition of

quasi-equilibrium and the LED operates in steady-state. In this steady state, current

flows in the direction of bias, electrical power is drawn from the power supply, some

light is emitted, the device heats up and loses heat to the environment, and the

net power entering the device through electrons, phonons, and photons reaches zero.

Although such low voltages are not commonly utilized, these operating points are

easily measured on existing devices, as seen in Figure 2-4.

-e

10-6

~10

10

1021-s -106 1e A

Current (A)-2

0.8

0.6

0.4

0.2a)

00 >0

Figure 2-4: I-V and L-I curves for an existing infrared LED emitting at A = 2 .15ptm.Current flows and easily detectable levels of light are emitted even when the appliedvoltage qV is far below the bandgap energy.

The steady-state condition is characterized by steady flows that obey continuity

equations. Since charge is conserved, a complete description of steady-state operation

52

Temp. Model Exper.25*C - o84*C -- - A

..3. ...C --- ---

obeys the following continuity equation at each point in space:

(-q)V.Je + qV.h = G-R (2.13)

where Je and Jh are the electron and hole fluxes, G is the local rate of electron-hole

pair generation, and R is the local rate of electron-hole recombination. A solution

may be visualized as in Figure 2-5.

S RecombinationiMins Genraion

Charge Flow

Figure 2-5: Charge flow in our simplified model obeys current continuity.

After considering the thermodynamics of thermally-assisted injection and recombi-

nation, entropy flow may also be considered. The Second Law permits the generation

of entropy, so the analogous continuity equation is:

V - is,e + V - is,h + V - S,lattice + V - JS,photon = S (2.14)

where JS,e, JS,h, J,lattice, and JS,photon are the entropy flux carried by the electrons,

holes, phonons, and photons respectively, and S is the irreversible entropy generation

rate. A solution may be visualized as in Figure 2-6. It is worth noting that such a

picture may be drawn for any electronic device, and that all inefficiencies in their op-

eration can be accounted for by some $, including those with significant consequence

for the engineering systems they compose.

53

PhotonField

Electronic ----- - -

Block Arrows DenoteEntropy Flow

PhononField r

Figure 2-6: Cartoon depicting entropy flux in a simple double hetero-junction LEDstructure.

2.2 The Heat Pump Picture

Let us now abstract away the internal dynamics of the electronic system and consider

just the flows of entropy and energy between the three sub-systems in Figure 2-6. For

each quantum of charge that flows through the device, one net recombination event

occurs. We would like to know how much entropy enters and leaves each system.

Knowledge of the energy flows between the sub-systems combined with Equation 2.11

and Equation 2.12 determines the entropy flows in and out of the lattice and photon

fields respectively. However, because the electronic sub-system is not in equilibrium

at any fixed temperature, we must examine it more closely.

We begin with a simple model for the electronic degrees of freedom at a single point

in space. Consider the statistical two-level system shown in Figure 2-7. Define fc to

be the probability of occupancy for the higher energy state, fv to be the occupancy

of the lower state, and take the states to be separated by energy AE. In terms of

these quantities then, we may write expressions for the total energy and entropy of

54

AE

Figure 2-7: A statistical two-level system.

the system:

U=fc-AE + (fc+fv).Eo and (2.15)

S = -kB [(f, In fc +1 (I - fc) ln(1 - fc)) + (fv - fc)] .(2.16)

If we define a degree of freedom corresponding to excitation from the lower state to

the upper state, we may find the amount of entropy change in the system per unit

energy change for changes of this type. This ratio can be expressed conveniently as

the inverse temperature T 1 of the electronic system:

dS dS

T- OS f dfv (2.17)OU U dU

dfc df,

dS= -kB [ln fc + 1 - ln(1 - fc) -1] (2.18)

dfc

-kB n ( c (2.19)

_ S -kB n ) + kB In ( (2

T - = -- = .-f (2.20)OU AE

If we constrain the probability for occupancy of either state fc + fv to be 1 so that

the Fermi level EF falls halfway between the states in energy, the equation above can

be rearranged to recover the expression for Fermi-Dirac occupancy in equilibrium at

55

temperature T:

exp ( EB) = 1f 1 = ( fc) 2 (2.21)(kBT I-fv fc fc

exp ( --/2 1 (2.22)

fc= (exp Eupperstate - EF -1 (2.23)

The preceding result is unsurprising, but clarifies an important point. The inverse

temperature of a Fermionic system, meaning the amount of entropy that is added to

it when a unit of energy is added, can be calculated purely from the occupation of the

states. That is to say, two situations which are described differently must still have

the same temperature if their occupancies are the same. To see how this applies to

the thermodynamics of electrons and holes in the active region of an LED, consider

the following slightly more concrete example.

-- - ---- - Ege--------- -Ege,EFh ~ AEFY= Ea--p --- EF,e, EF~h

- - - - - - - - - EF~h

Tlatte = 300K r = 300K Tttie = 300K T* = 600K Tttice = 600K T* = 600K

(a) Two-level system in (b) Two-level system ex- (c) Two-level system ex-equilibrium. cited electrically. cited thermally.

Figure 2-8: Two-level systems that exhibit different types of excitations which lead tothe same occupation of states have the same effective temperature T*. In Figure 2-8a,the electronic system is in equilibrium with a 300K lattice. In Figure 2-8b, theelectronic system is not in equilibrium with the lattice. A Fermi-level separation hasincreased the occupancy of the higher-energy state and decreased the occupancy of thelower-energy state. Although the lattice temperature in Figure 2-8b is still 300K, theeffective temperature T* that indicates the ratio of entropy to energy in the electronicsystem is 600K. In Figure 2-8c, the electronic system is again in equilibrium with thelattice, but the lattice is now at 600K. The occupancies fc and fv in Figure 2-8b andFigure 2-8c are identical, so their values of T* are the same.

56

Consider an ensemble of homogeneous quantum dots, each with a single relevant

low-energy electron state and a single relevant high-energy state. Again let the total

charge between the states be fc + fv = 1 to ensure charge neutrality. If the lattice

of these dots is kept at 300K and no electrical excitation is applied, the statistical

two-level system will have a Fermi level at exactly halfway between the two states

and the occupancies fc and fv can be determined by the Fermi-Dirac distribution.

This situation is described by the diagram in Figure 2-8a.

Now let us excite this system. Since a recombination event removes an electron

from a higher energy state and places it in a lower energy state (and vice versa for

a generation event), let us again focus on the degree of freedom corresponding to

fc -+ fc + 6f and fv -+ fv - 6f. Note that this is the same degree of freedom that

we used in Equation 2.17 and corresponds to excitations that conserve total charge.

Figure 2-8b and Figure 2-8c show two physically different types of excitations that

result in the same values of f, and fv. In Figure 2-8b, the electrical system has

been taken out of equilibrium with the lattice by an applied voltage qV = AE/2.

In Figure 2-8c, the absolute temperature of the lattice has been doubled. In both

situations, the number of kBT'S of between each state and its quasi-Fermi level has

been halved. As a result, the Fermi-Dirac occupation of the states in both situations is

equivalent (i.e. fc and fv are the same in both). Since the total entropy S and energy

U of the system is determined solely by fc and fv, the procedure from Equation 2.17

yields the same temperature T = (S/U) 1 for either situation. From now on we

will refer to this temperature as the effective temperature T* seen by the inter-band

processes like radiative recombination.

From these examples, we may follow [43] to a general expression for T* in a

semiconductor whose quasi-Fermi levels are separated by an energy qV in a region

with bandgap energy Egap:

(V_-_T* = Tiattice 1 _E . (2.24)

egap

From here, we may significantly simplify the internal dynamics of the electronic

57

system from a picture like Figure 2-6. For inter-band processes in which the electronic

system loses energy to another reservoir (e.g. recombination), the corresponding loss

of entropy is determined by T* from Equation 2.24.

By contrast, for intra-band processes like thermally-assisted injection, the two-

level system model is not necessary. For the 3D semiconductors in the simple LED

model we will use going forward, the distribution of carriers at a given position and

within a given band is approximately thermal. At a position in the device where

a forward bias causes the carriers in a specific band to flow "uphill" toward higher

electrostatic potential energy, the injection process involves an inflow of carriers at

low energy and an outflow of carriers at high energy. As described in § 2.1.3, in

steady-state the energy absorbed via Peltier heat exchange with the lattice supplies

the energy for the re-thermalization of these carriers by moving carriers from more

occupied low-energy states into less-occupied higher-energy states. As they do so,

the carriers move from portions of phase space which are more densely populated

to portions that are more sparsely populated. This flow of carriers thus increases

the number of microscopic configurations of the electronic states at this position;

the carriers thus absorb entropy along with energy from the lattice. The amount of

entropy absorbed is determined by the same temperature that determines the spread

of carriers in phase space in that location within that single band. As a result,

for intra-band processes in which the electronic system gains energy from another

reservoir (e.g. thermally-assisted injection), the corresponding amount of entropy

added to the electronic system is given by the local lattice temperature Tiattice-

If we modify Figure 2-6 by consolidating all flows of entropy together, and we also

include the corresponding flows of energy from the various sources, the picture be-

comes the canonical diagram for a thermodynamic heat pump as shown in Figure 2-9.

2.3 LEDs in the Low-Bias Regime

As described in Chapter 1, it has long been known that at low output power an LED

may in principle operate with wall-plug efficiency q; far in excess of unity [29, 31].

58

Photon Field

IrreversibleEntropy

Generation

Non-radlativeRecombination

Phonon Field Phonon Field

Figure 2-9: The flows of entropy and energy between various sub-systems in an LEDcan be organized in the canonical picture of a thermodynamic heat pump. At left isan idealized picture. The irreversible contributions shown in the picture at right canbe quantified for any real LED using the arguments from § 2.2.

That is, its optical output power (L, measured in Watts) may be a large multiple of

its input electrical power (IV, also measured in Watts) in steady-state. In fact, the

Second Law of Thermodynamics permits an arbitrarily large value of 7. This is the

situation in the low-bias regime we will discuss here.

We pause briefly to address a question of terminology. Typically the ratio of the

rate at which heat (in this case, photons) is emitted by a heat pump to the rate

at which it consumes work is called the pump's heating coefficient of performance

COPH, but in this work we refer to this quantity as the wall-plug efficiency q. We

note that in other electrically-driven sources of incoherent light for which q < 1, the

output energy also has entropy associated with it, so that L/(IV) would be most

appropriately termed COPH in this case as well. Nevertheless, convention dictates

that L/(IV) is referred to as the wall-plug efficiency q. For this reason, we follow

several previous authors [31, 29, 14, 351 in referring to this quantity as the wall-plug

efficiency (or simply efficiency) q, which we allow to exceed unity.

Recall the expression for the wall-plug efficiency of an LED from § 1.1:

(OW)q= 7 7EQE - (2.25)qV

59

Although different device structures and material systems lead to various types

of recombination whose rates (both relative and absolute) can vary widely, here we

will consider three processes: trap-assisted Shockley-Reed-Hall recombination, bi-

molecular radiative recombination, and Auger recombination. The rates of SRH,

bimolecular, and Auger recombination are typically expressed in terms of the elec-

tron and hole concentrations, n and p respectively, while all other dependences are

captured by some phenomenological rate constant (here A, B, and C). It is worth

noting that these constants are intended to be independent of the magnitude of the

local electrical excitation; the n and p dependences capture that physics. The most

common form of these expressions appears below:

(np - ni)(n + n r)T + (p + pi)(r.

A(n - no) or A(p - po) (2.27)

Rrad = B (np - n 2) (2.28)

RAuger = C (n (np - ni2) + (np - n )p) (2.29)

Instead of the carrier concentrations n and p, these rates can be rewritten in terms of

the Fermi level separation, taken to be equal to the applied voltage qV. In the dilute

Boltzmann limit, the product np rises exponentially as with qV so that

np = ni (eevkBT) (2.30)

Where doping is used to create a large majority carrier population at equilibrium,

the increase in the product np in response to a small forward bias is due to the

increased minority carrier density. That is to say, the quasi-Fermi level of the majority

species is relatively fixed while the quasi-Fermi level of the minority species is moved

closer to the minority band edge, increasing that carrier density. Thus, to a good

approximation

P = Po and n = no (eqV/kBT where p > n and (2.31)

60

n = no and p = po (eqv/kBT) where n > p. (2.32)

Substituting these expressions into Equation 2.26, where p > n we have

n2 (eqv/kBT _ n (eqv/kBT -

RSRil - 2 2-33)(Po + p1)rn 2T L Tnno

= Ano (eqv/kBT - 1) (2.34)

Rrad = B ni (eqV/kBT - i) (2.35)

RAuger = Cpo n2 (eqv/kBT (2.36)

where we have assumed the states contributing to SRH recombination are near the

zero-bias equilibrium Fermi level and the trap lifetimes mr and Tp were on the same

order. A similar expression can be derived for n > p. At some point along the

junction, n is on the order of p. Here we can write simple expressions for n and p

which are valid when qV/kBT < 1 in terms of the carrier asymmetry x = no/(no+po).

P = Po (exqV/kBT) and n = no (e(l-x)qv/kBT) where p - n. (2.37)

Note that for larger voltages, the effects of carrier asymmetries will wash out in the

same way as doped regions experience at much higher bias. Beyond this point, if

both species remain in the Boltzmann limit, both Fermi levels move toward their

respective band edges symmetrically (i.e. n and p grow with qV like when x = 1/2

in Equation 2.37). Substituting as before, for regions with p ~ n, we have:

2i (eqvlkBTRSRH (v/B - 1 (2.38)

(no (e(1-x)qV/kBT) + po (exqV/kBT) + 2ni) - T

Rrad = B n (eqv/kBT - 1) (2.39)

RAuger = C - [no (e(1x)qv/kBT) ±po (exqv/kB) (V/kBT - 1) . (2.40)

Now let us imagine a device in which the active region extends from x = 0 to

x = L and consider three separate regions: p > n over (0,Xp), p ~ n over (Xp,Xn),

and n > p over (XnL). From the expression for EQE from Equation 1.1, if we

61

assume the extraction and injection efficiencies to be independent of applied bias, we

can capture the voltage dependence of the quantum efficiency:

77EQE C)C (Rradiative)active (2.41)(RSRH)active ± (Rradiative)active ± (RAuger)active

From the above equations for RSRH, Rrad, and RAuger, it is clear that all three recom-

bination processes have nonzero contributions at linear order in qV/kBT.

This may at first seem counter-intuitive, because we typically think of defect-based

SRH recombination as a one-particle process, radiative bimolecular recombination as

a two-particle process, and non-radiative Auger recombination as a three-particle

process. While this is true, not all of the particles in these processes need to be excess

particles. Some can be thermally-generated equilibrium carriers that exist when the

device is off but at finite temperature. In fact, if we were to ignore the thermally-

generated equilibrium carriers, we should not expect q > 1 operation to be possible,

since the low-temperature reservoir would be at T = OK and have no entropy.

The fact that radiative bimolecular recombination has a finite contribution at

linear order in the dimensionless electrical excitation qV/kBT implies that the external

quantum efficiency of a very general class of LEDs remains a nonzero constant as

V -+ 0:

lim 7EQE # 0 (2.42)V-+O

Experimental evidence of this behavior is presented primarily in Chapter 3, but

the basic fact is readily apparent in Figure 2-10. From this it follows that arbitrarily

high wall plug efficiency is achievable at low voltage:

lim 'q = lim O 7EQE = 00 (2.43)V -+0 v--+o qV

This type of behavior, where unbounded coefficient of performance for heat pumping is

available at arbitrarily low power is a general feature of thermodynamic heat engines.

We discuss this trade-off futher in § 2.4.3.

62

kIT/q @ 135"C

IkT/q @ 84"C

- 15 H Ed H H -

. a-m- z W-O

840C k T/q @ 25-C

----j ---± --

104 10-2Voltage (V)

-P

*0

/ 2

10

10~ 100

Figure 2-10: The quantum efficiency of a conventional LED approaches a constant asthe applied voltage falls below kBT/q (~ 25 meV at room temperature). The discretemarkers represent experimental data while the lines represent simulation results basedon the equations presented in this chapter.

63

10

10

10 -

1350C00G)

Lii0uJ

10- . . . A

10 -

2.4 Carnot-Efficient LEDs and Real LEDs

2.4.1 Carnot Efficiency

I

OperatingPoint

Figure 2-11: I-V curve for an ideal LED. Input electrical power is represented by

the red box between the origin an the point (V,I) while output power is represented

by the larger box between the origin and (hw/q,I). As the operating point moves to

lower voltage, the ratio of these areas (i.e. the wall-plug efficiency) diverges.

Consider the I-V curve of an LED with unity quantum efficiency as shown in

Figure 2-11. As usual, the electrical input power into the diode is given by IV. Here,

this quantity is represented by a box between the origin and the operating point

(V,I). Now, since this LED has unity quantum efficiency, the rate at which photons

exit the device is equal to the rate at which charge flows through it, and each photon

carries away hw worth of energy, the output power is represented by a box between

the origin and the point (hw/q,I). From this picture, several simple features can be

seen.

64

2 CO

First, for any forward bias voltage V < hw, output power exceeds input power.

This means the device is cooling. Subtracting the box corresponding to input power

from the box corresponding to output power gives the cooling power. Also, since all

these boxes are the same height, the ratio of output to input power q can be easily

visualized as hw/qV. Finally, we can see that this ratio diverges as V becomes small.

As we will discuss shortly in § 2.4.3, it is also apparent that as this happens, the

amount of current flowing is also reduced and the power flowing through the system

becomes small as well.

Recall now that in § 1.1 and § 1.2, we examined two simple but very different

expressions for the efficiency of an LED. The first expressed that each electron that

flowed through the device could result in the emission of a photon of energy hw with

probability EQE. Since qV of electrical energy is required to drive this current, we

wrote:

w= - EQE - (2.44)qV

Then in § 1.2, the maximum efficiency permitted by The Second Law was expressed in

terms of the lattice temperature of the device Tattice and the temperature of outgoing

photon field Tphtn..:

7 1Carnot Thoton (.) (2.45)Tphoton (I) - Tattice

As we will see shortly, these two expressions lead to a singular concept of an ideal,

Carnot-efficient LED.

Returning to the ideal LED whose I-V curve is shown in Figure 2-11, let us

determine the output power at an operating voltage V. When a voltage V is applied,

the quasi-Fermi levels of the active region separate by an energy AEF = qV. The

conduction band states are then occupied with more electrons than at equilibrium;

the valence band states also contain more holes. Recalling the logic from Figure 2-8

and Equation 2.24, the occupation of these states is roughly equal to the occupation

at equilibrium at the elevated temperature T* = Tattice(1 - (qV/hw))-. Thus we

might expect the output power to match the spectral intensity of a blackbody at T*.

65

If the active region of the device is many absorption-lengths thick at the emission

wavelength, it should radiate with unity emissivity.

If the intensity I from an ideal device is just the blackbody intensity at temper-

ature T* over the relevant spectrum, then Tphoton(I) = T* in the expression for the

Carnot efficiency. Substituting the expression for T* into Equation 2.45 gives:

Warnot Tphoton (Iideal) -(.6U'Carnot = (2.46)Tphoton (lideal) - Tattice

T* (2.47)

T* - Tattice

T attice ( r - ' (2.48)

Tattice (1 - ) - Tattice

77 (2.49)qV ideal

2.4.2 Non-Ideality of Existing LEDs

For real LEDs, of course, the effects of non-radiative recombination are substantial

and EQE < 1. Although the descriptions of these devices can be quite complex, the

relationship between them and a Carnot-efficient device is captured entirely by IJEQE

when the voltage is well below V = Egap/q and the active region carriers are in the

Boltzmann limit.

At higher voltages corresponding to conventional operating points, diodes can

reach transparency and inversion, so that both our approximation of an optically-

thick active region and the Boltzmann approximation become invalid. If the electronic

system is inverted, for example, then even recombination events that result in a final

photon state with no entropy (i.e. lasing) can be thermodynamically preferred. If we

naively applied the equations above to such a situation, we would predict an infinite

photon flux corresponding to an infinite brightness temperature as V -+ Egap/q. Since

this is neither physical nor in agreement with observations, we should expect that at

some point the radiative transition becomes saturated and the intensity falls below

that of the effective temperature T* from Equation 2.24.

66

2.4.3 The Power-Efficiency Trade-Off

Even a Carnot-efficient LED faces a fundamental constraint on its spectral intensity

due to the finite phase-space density of photon modes and the speed of light. As de-

scribed in @ 1.2, a given spectral intensity of a light source, 1(A), requires a particular

minimum temperature Tph ot o n of the outgoing photon field.

Itotal (A) = 2whc2 A 5 (2.50)exp AkB photon

'background (A) = 2hc 2 A 5 (2.51)exp ABT..bient

L ' Itotal - 'background Oc exp h exp [ e 1 (2.52)exp - 1 onlp AkBTambientI )

Meanwhile, the outgoing photon field temperature Tphoton limits the efficiency by the

Second Law:

7 Carnot = (2.53)Tphoton - Tambient

For a given wavelength and lattice temperature, a fundamental connection can

therefore be made between power and efficiency. In this way, the Carnot efficient

LED is analogous to other endo-reversible heat engines, but with its finite thermal

conductance from the electron-hole system of the active region to the photon field set

by the Planck radiation law. This trade-off between power and efficiency is depicted

for various wavelengths of interest in Figure 2-12.

A regime of special interest may be found when AT = Tphoton -Tambient < Tambient-

67

10

10

103 0 0

102

0

C)

>-10LU

10 -15 -10 -5 010 10 10 10

Spectral Intensity (W/m 2/nm)

Figure 2-12: Efficiency versus spectral intensity of the electrically-driven optical powerfor Carnot-efficient LEDs emitting at various wavelengths of interest. From left toright they are 555 nm (peak response of the human eye), 1104 nm (Silicon absorptionedge at 300 K), 1550 nm (SiO 2 fiber loss minimum), and 2600 nm (emission wave-length from experiment in § 3.2). At all wavelengths, there is a low-power regimein which the outgoing optical field is barely brighter than the blackbody backgroundand efficiency scales inversely with power. For longer wavelengths, this correspondsto a higher intensity. Note: calculations assume ambient temperature of 273K.

68

Defining Tphoton/Tambient =1 + x and expanding for small x, we see:

L oc I(2.54)

exp AkBTarnbint(1+X kBTambien

L oc

exp B ambientl B amintBabin

(2.55)

[x he hex

L oc hc arnbient B arnbient oc (2.56)exp Ak h -

Thus, since

WCarnot = = = ±- + 1 (2.57)Tphoton - Tambient (1 + X) - I X

small x corresponds to high efficiency q > 1, where Warnot oc 1/L. This behavior

can be seen readily in Figure 2-12. At each wavelength, below some power level the

slope of the Carnot bound becomes 1/L. Because infrared wavelengths carry more

blackbody radiation at typical ambient temperatures, this transition occurs at higher

power for these wavelengths than visible wavelengths. As we will see in Chapter 3,

this will lead us to focus experimental efforts on infrared emitters.

2.5 Design of LEDs for Heat Pumping

As we saw in § 2.4.2, although existing LEDs share certain qualitative features with

ideal LEDs, non-radiative recombination, leakage, and imperfect photon extraction

act as significant sources of irreversibility and cause real LEDs to operate far from the

Carnot efficiency bound. This is particularly true for infrared LEDs, for which lower

material quality in the active region leads to shorter trap-assisted non-radiative life-

times and smaller bandgap energies lead to increased Auger recombination and carrier

leakage. In order to redesign devices with improved optical power and conversion ef-

ficiency at low voltage, Dodd Joseph Gray, Jr. and I have created and experimentally

validated a numerical model of charge and heat transport using the commercial soft-

69

ware package Sentaurus Device distributed by Synopsys. The material parameters

in this simulation (and the simulations used to fit experimental data in § 3.2) were

modeled with the equations in Table 2.1 which reference constants in Table 2.2 and

Table 2.3. Starting with the structure of an existing Gao.s5 Ino.15Aso.13 Sbo. 8 7/GaSb

2.15pm LED designed for high-bias room temperature operation, we alter the active

layer thickness, active layer doping, operating temperature, and active material SRH

lifetime to improve on this design at low-bias. The results reported here mirror those

reported in Ref. [69].

Model Formula

Bandgap Egap = Egap,o - #(T - 300) - AjNj 3 - BzN - C- N /

Density Ni =NO T 3/2

of States 300

Mobility i= Pi 0 )300

SRH RSRHnp - niRecombination TSRH,h,O 3/2 (n+ni) +- TSRH,e,O (300 (p ni)

Surface SRH np - n2Recombination RSRH,surf = Vsurf(n + p + 2ni)

Bimolecular Rr = B (T)-3/2 gap (np - n?) ,whereRecombination

B = BO f (a) = BO x 0.15 (See pp. 67-74 of Ref. [9])

Auger Ru,=C( )(p-n2Recombination RAuger = C(n ip - r )

Table 2.1: Equations describing phenomenology of material parameters. The con-stants which were used with these equations are found in Table 2.2 and Table 2.3. Inthese tables, T refers to the absolute temperature in Kelvin and when i appears as asubscript of a capital letter it stands in for the carrier species or dopant type.

70

Parameter Name SymbolFor GaInAsSb(85% Ga, LM) For GaSb

Intrinsic bandgap Egapo 0.583 eV [70] 0.726 eV [71]

Thermal bandgapnarrowing 3.78 x10- 4 eV K-' [71] 3.78 x10- 4 eV K- 1 [71]parameter

Jain-Roulston n-type An 1.36x10-8 eV-cm [72]bandgap narrowing Bn Same as 1.66x10- 7 eV.cm 3/ 4

parameters Cfor GaSb 1.19x10-1 0 eV.cm 3/ 2

Jain-Roulston p-type ASame as 8.07x10-9 eV-cm [72]bandgap narrowing Bp for as 2.80 x 10-7 eV-cm 3/ 4

parameters CP for GaSb 4.12x10- 1 2 eV-cm 3/2

Electron SRH lifetime TSRIH,e,O variable iOns [73]

Hole SRH lifetime TSPH,h,O variable 600ns [73]

Surface SRHrecomb. velocity Vsr 1900 cm/s [74] 1900 cm/s [74]

Radiative constant BO 3x10"1 cm 3/s [75] 8.5x10- 11 cm 3/s [73]

Auger constant C 2.3x10- 28 cm 6/s [74] 5x10- 30 cm 6 /s [73]

Absorption a 4000 cm- 1 [74] Not Used

Electron mobility pe,O 5000 cm 2 /Vs [76] 3150 cm 2 /Vs [73]

Hole mobility ph,o 850 cm 2 /Vs [77] 640 cm 2 /Vs [73]

Electron mobilitytemp. exponent 'e 1.9 [78 0.9 [79

Hole mobilitytemp. exponent 2.3 [78] 1.5 [73]

Conduction banddensity of states Nco 1.9x10 17 cm- 3 [80, 81] 2.1x10 1 7 cm 3 [81]

Valence banddensity of states N,, 1.5x10 19 cm 3 [80, 81] 1.8x10 1 9 cm 3 [81]

Table 2.2: Material parameters associated with the electrons and holes. Values are formaterial at 300K unless otherwise specified. Note that the figure given for absorptionce refers to the value approximately kBT above the absorption edge.

71

280 300 320 340 360 380Temperature (K)

400 420 440 460

280 300 320 340 360 380 400 420Temperature (K)

440 460

Figure 2-13: Output optical power density at unity wall-plug efficiency L,=1 versus

operating temperature. At top, L7= 1 is plotted for three n-type dopant densities

of ND = 3 x 10 1 cm-3 (blue dashed line), ND = 2 x 10 16 cm- 3 (red solid line) andND = 6 x 10 17 cm- 3 (black dot-dashed line), demonstrating that an optimal dopant

density exits for low bias operation at 300K. Hollow squares denote experimental data

from Chapter 3. At bottom, we plot L.= 1 as a function of temperature for n- (red)

and p-type (blue) doping. Solid lines denote calculations with the GaInAsSb SRH

lifetime T = 95ns and dashed lines denote T = lys.

72

1

108

~10

_j

1- - 1

Experiments (undoped)

NND2 X 101 7 m

- -4 - 3 N1 =6 X 10cm1 -'P 010 000ND 3 X 1014 Cm-3

TSRI =1S 95ns

p-type --

.. *SRM 'I'S 95ns

n-type

E.

E

_j

106

10 8

10-10

1012

10-1

For GaInAsSbParameter Name Symbol Fo% Ga, LM) For GaSb(85% Ga, LM)

Lattice ThermalCodtivit Thermalattice 14 W/mK [82] 3 W/mK [82]Conductivity

Static Dielectric 15.64 [80,81] 15.7 [81]Constant

Series Rseries 0.779 Q (fit in Ref. [47])Resistance

LightCollection 7collection 24.5% (fit in Ref. [47])Efficiency

Table 2.3: Remaining material parameters not included in Table 2.2.

In Figure 2-13 we compare the power density at unity wall-plug efficiency versus

temperature across various designs of a 2.15pm LED. At top the plot compares de-

vices with differing levels of n-type doping in the active region. Near 300K, designs

with ND -2 x 10 16 cm- result in more than a 10x improvement over both nominally-

undoped and heavily-doped designs. Results across the range of dopant densities and

temperatures (not plotted here) point to the existence of an optimal doping concentra-

tion at this temperature. At higher temperatures, the intrinsic carrier concentration

ni is higher so that the importance of doping is diminished. Above about 400K, the

optimal dopant concentration is small so the original structure is nearly optimal. At

these temperatures, experimental data from Chapter 3 matches our numerical results

closely.

Intuitively, doping improves device operation at low bias by increasing the internal

quantum efficiency [14, 47]. In the low bias regime at low temperature, excess minority

carriers experience non-radiative trap-based SRH recombination as well as radiative

bimolecular recombination. Since the rate of bimolecular recombination is linear in

the concentrations of both electrons and holes, for a given minority carrier density, a

doped structure with greater majority carrier density will experience more bimolecular

recombination. Meanwhile, the SRH recombination rate is linear in the minority

73

carrier density and the trap density, but is nearly unchanged with majority carrier

concentration (i.e. doping). Increasing the bimolecular recombination rate which

determines optical output power relative to the typically dominant SRH process leads

to an increase in quantum efficiency. This explains the initial increase in power at

unity wall-plug efficiency with dopant concentration.

At very high doping, Auger recombination becomes relevant even for voltages be-

low the thermal voltage, and the linear increase in bimolecular recombination rate is

outweighed by the quadratic increase in Auger recombination with dopant concen-

tration. For example, when CCCH-type Auger is the dominant Auger process, the

Auger recombination rate may be expressed as RAuger C rn2p. In an n-type material,

this leads to the aforementioned quadratic dependence on doping. If the material is

highly p-doped, other processes like CHHS or CHHL replace the CCCH process in

the logic above, but the quadratic dependence remains. Combining this general result

with the previous result relating bimolecular and SRH recombination rates, we find

there exists an intermediate dopant concentration at which the quantum efficiency at

low voltage is optimized. In keeping with the result given by Heikkila, et. al. in Ref.

[14], the low-bias quantum efficiency is optimized when the dopant concentration is

V7C, where T is the SRH lifetime and C is the Auger coefficient.

The preceding analysis was done assuming an excitation characterized by a con-

stant excess minority carrier population. However, as doping is changed, the forward

bias voltage corresponding to this density changes. If we translate the logic above

into terms at constant voltage, we find that doping the active region suppresses SRH

recombination by reducing the equilibrium minority carrier population while the bi-

molecular recombination rate, which is proportional to n?, is unchanged due to the

law of mass action (i.e. nopo = n2). The Auger recombination rate is increased with

doping because the quadratic dependence on doping described above is not com-

pletely offset by the decreased equilibrium minority carrier population. Thus we find

that the same conclusions hold. The low-bias quantum efficiency and power density

at unity efficiency are maximized at a finite optimal dopant concentration which re-

flects a balance between the parasitic SRH and Auger non-radiative recombination

74

pathways.

The lower plot of Figure 2-13 shows the power density at unity wall-plug efficiency

versus temperature for four different devices. The two solid curves correspond to

devices with active region SRH lifetime 7 = 95ns while the dotted curves correspond

to T = 1[ts. The devices with longer active region SRH lifetime have higher quantum

efficiency and thus higher L, 1 . The blue curves indicate the results of simulating

structures with p-type optimally doped active regions while the red curves correspond

to n-type doping. The p-doped devices have almost an order of magnitude higher L. 1

at a given temperature as their n-type counterparts. This asymmetry can be explained

by the differences in SRH recombination rates near the hetero-junction between the

n-GaSb region and the intrinsic InGaAsSb active region. At this interface, a narrow

region exists in which the electron density is very high, due to the difference in

electron affinity between GaSb (4.06eV) and the quaternary alloy (4.18eV) in our

model. Excess electrons in this region can experience SRH recombination or undergo

spatially indirect transitions to valence states at adjacent locations where the hole

density is also high. A more thorough analysis including experimental data for band

alignments would likely enhance the predictive power of this model, but as with many

facets of simulations in the InGaAsSb material system, conclusive experimental data

remains scarce. As a result, predictions about the relative values of quantum efficiency

for n-doped and p-doped active region designs are less firm than other predictions like

the existence and magnitude of an optimal doping level.

Figure 2-14 presents a breakdown of the various recombination processes con-

tributing to conduction through an optimally p-doped diode as a function of temper-

ature. At all temperatures, SRH recombination in the active region is the dominant

pathway. At higher temperature, the relative strengths of the active region processes

increase for the multi-particle Auger process while the relative strength of the one-

particle SRH process decreases. This is in keeping with the explanation provided

above. Furthermore, higher temperature leads to an increase in leakage current. This

is to be expected, since as temperature increases, there is an exponential increase in

the fraction of carriers able to thermionically emit over the hetero-barriers and escape

75

SO"

.2E00

0a)

1014

1012

1010

300 350 400Temperature (K)

450

Figure 2-14: Radiative (red solid line), SRH (black dashed line), and Auger (blackdot-dashed line) recombination rates per unit area in an optimally p-doped structure

plotted as a function of device lattice temperature. The leakage curve (blue dashed

line) combines all recombination processes outside the active region and may be seen

as a parasitic component of the current density flowing in response to an applied

voltage. The data shown here represents the results of a diode with active region

SRH lifetime r = yIps at the unity efficiency operating point.

76

,,.--eaka -

SRH in Active Leakage,

- Auger in Active

Radiative in Active

-0000

I

I

the active region to undergo recombination in the quasi-neutral regions or near the

contacts.

1.5E

E0

.= 0.5

11

x 10-1

1 2Thickness (gm)

3 4

Figure 2-15: Power at unity wall-plug efficiency versus active region thickness forstructures with three different levels of p-type doping. The inset is a plot of theextraction efficiency versus thickness, which is a decaying exponential due to reab-sorption of photons generated within the active region. At all three doping levels, wefind an optimal thickness which reflects the trade-off between reabsorption and theneed for substantial active region volume to outweigh the effects of leakage.

The results of varying the active region thickness of these structures is shown in

Figure 2-15. For each of the three p-type doping levels examined, as well as others

not shown here, there exists an optimal thickness which maximizes the power at

unity efficiency. At very small thicknesses, the total fraction of current which passes

through a recombination pathway in the active region is proportional to thickness.

In essence, a thicker active region diminishes the importance of leakage current by

increasing the total current, and thereby increases the quantum efficiency and thus

Lni. At large thicknesses, the majority of photons generated through radiative

77

NA 2 X 1ol cm-3 NA= 3 X 10 cm-3

61017

0.5

NA= 6 X 1017 CM-3 Thickness (pm)IIIn0

recombination undergo reabsorption before they can escape the active region. The

probability of escape, or equivalently the extraction efficiency 77ext, was calculated

from a 3-dimensional model assuming a uniform distribution of photon generation

in position and angle and a distribution in energy proportional to the density of

electron-hole pairs connected by vertical transitions at low bias at 300K [9]. The

results of this calculation differ only by a constant factor from what one would expect

from a one-dimensional model, namely an exponential decay of xext with thickness.

Given this extraction efficiency, since the fraction of active region recombination that

is radiative (i.e. the ratio of recombination rates expressed in Equation 1.1) is small,

the external quantum efficiency also drops exponentially with thickness. Combining

these two mechanisms results in curves of the shape seen in Figure 2-15, with an

optimum active region thickness around 1.5pm.

Figure 2-16 shows the results of the final redesign. This design includes optimal

p-type doping and optimal active region thickness, both chosen to maximize L.= at

298K. The Carnot limit shown here is calculated by combining the experimentally-

measured spectrum at 300K with a given power density to find the brightness tem-

perature for the average photon Tphoton, which is subsequently used in the well-known

expression for the maximum efficiency of a heat pump from Equation 2.45. The

redesigned structure's overall behavior of achievable efficiency versus power density

resembles the Carnot efficiency more closely than the original design. Since the re-

maining difference is directly connected to the external quantum efficiency, which is

limited primarily by non-radiative SRH recombination in the active region, the de-

sign is likely within a few percent of optimality given the assumed SRH lifetime and

bimolecular recombination coefficient.

2.6 Circuits are Cycles

When a battery is connected to a diode, current flows in a loop. Electrons flow from

the negative terminal of the battery through a wire to the device's cathode, across

the device from cathode to anode, back to the battery's positive terminal, and finally

78

ai)

0)

100

1

.011

f~~~4I

1 -11 -1 -10 -9 -8 -7 -6 -5 -4 -310 0 10 10 10 10 10 10~ 10 10

Power Density (W/mm2)

Figure 2-16: Results of the redesign of the 2 .15pm LED for low bias. The plot showswall-plug efficiency as a function of power density for the existing device characterizedin § 3.2 (dotted red line represents simulations while the discrete markers representexperimental data), as well as the simulation results from the redesigned device (solidblue line). Also included is an estimate of the Carnot limit (black dashed line) forwall-plug efficiency as a function of power density. The redesigned device has orders ofmagnitude better performance at brightnesses on the nanowatt per square mm level.The overall behavior resembles the Carnot efficiency more closely than the originaldesign, but is still limited by non-radiative SRH recombination.

79

Redesign

E D

Existing Device -( ee

back through the battery to its negative terminal. At each point along this path, a

given electron experiences a range of environments. Although these environments are

typically described using various transport frameworks based primarily on statistical

mechanics, they may also be described by thermodynamic state functions. When

described in this way, the simple battery-diode circuit is analogous to an internal

combustion engine as described in Figure 2-17.

Internal Combustion Engine

S3 @@@(D@@

Figure 2-17: The path of electrons through a circuit is a closed loop and may bedescribed by a thermodynamic cycle. Along this path electrons may exchange energyand entropy with other reservoirs such phonons or photons just as the working fluid ina more conventional thermodynamic machine may exchange energy and entropy witha condenser plate, heat sink, or mechanical subsystem with few degrees of freedom.The electrons in different parts of the circuit at left are at different stages in the samecycle. The circuit is analogous to the internal combustion engine at right in thatdifferent portions of the working fluid are at different phases of the same cycle.

The most common descriptions of circuits are based on statistical mechanics.

Although a macroscopic circuit involves many microscopic degrees of freedom, we

typically care only about the aggregated similarities among the collection of degrees

of freedom. For example, when current flows through a wire, we care primarily about

the resistance of the wire. That resistance is a measure of the average momentum

of electrons down a small electro-chemical potential (i.e. Fermi level) gradient. The

current does not care about what distribution of electron momenta give rise to that

average.

The descriptions we have offered in this chapter are different. By considering the

flow of entropy within an electronic device, we are asking not about the similarities

80

between the dynamics of the microscopic degrees of freedom, but about their dif-

ferences. In this chapter we have focused on the flows of entropy between different

sub-systems through interactions (e.g. entropy flow from the lattice to the electrons

in thermally-assisted injection). We found this most useful because the purpose of an

LED is to transport energy from one domain to another. For other types of devices,

mapping the flow of entropy within a given sub-system (e.g. within the lattice, or

within the band-edge states of relevance to charge transport) may prove useful. More-

over, since any closed circuit is also a closed cycle for the electrons that flow through

it, in addition to single devices like transistors, even complex integrated circuits may

be amenable to thermodynamic analysis of this type.

By modeling entropy flow directly, we can directly identify the origin of any irre-

versible entropy generation which must underly any differences between real electronic

machines and their idealized conceptions. In much the same way as the thermody-

namic analysis of combustion engines allowed the development of new cycles and new

engines based on them, it seems plausible that a systematic study of the entropy

flow in electronic devices could yield practical design improvements. We discuss this

subject again in Chapter 6.

2.7 Summary and Conclusions

In this chapter we have assembled a theoretical framework for the thermodynamic

analysis of transport in a semiconductor light-emitting diode. We began with a

detailed description of the entropy flows involved in the basic electron transport pro-

cesses in an LED, with emphasis on the case when the applied forward bias voltage

is less than the bandgap energy and both the electron and hole populations in the

active region are in the dilute Boltzmann limit. In this case, we found that the elec-

trons absorb entropy from the lattice during injection and release entropy with the

outgoing photons. We recognized that this behavior is in close analogy with con-

ventional mechanical heat pumps, with the electrons acting as a working fluid to

transport entropy and energy from one reservoir (the lattice phonon bath) to another

81

(the outgoing photon field).

Given these tools, we were able to generate a heat pump diagram from the con-

ventional description of electron transport in a semiconductor device which identifies

carriers as flowing across an band diagram while experiencing generation and recom-

bination at various points. Next we considered what an ideal Carnot-efficient LED

would look like, and found that such a device corresponds to the case of perfect ex-

ternal quantum efficiency. We then saw that even such ideal devices would face a

fundamental trade-off between efficiency and power density, but that the constraint

was less strict for LEDs emitting at longer wavelengths. The latter observation will

serve as the motivating factor behind the experimental design in the next chapter.

We ended our theoretical discussion by applying the framework we built to re-

design an existing 2.15pm LED for more efficient electrical-to-optical power conver-

sion at low forward bias voltages. Our simulation results, which were based on an

experimentally validated model, indicate that existing growth capabilities are suf-

ficient to realize unity wall-plug efficiency at room temperature in an LED at this

wavelength. We closed our theoretical discussion by arguing briefly that the space of

problems amenable to this type of thermodynamic analysis is quite wide and in prin-

ciple includes any electronic device or combination of devices which forms a closed

circuit and operates in steady-state.

82

Chapter 3

Experiments on Existing Emitters

Although the prospect of very high efficiency (i.e. q >> 1) electro-luminescence ex-

pressed in § 2.4.3 was theoretically predicted by Jan Tauc as early as 1957 [31], the

phenomenon of q > 1 photon generation had remained experimentally unconfirmed

until the present work. In this chapter, we present a series of experimental mea-

surements of electrically-driven light emission from devices which were designed and

fabricated outside the scope of this work. These devices, mostly infrared LEDs, were

designed for conventional high-current (A/cm2 -scale and above) operation at room

temperature. By investigating their performance at high temperatures and small

currents, new physics was observed.

In addition to the first experimental evidence of above-unity electrical-to-optical

power conversion, this series of experiments provides empirical evidence to corroborate

the theory from Chapter 2. To confirm that 17EQE becomes voltage-independent in this

regime and that q therefore scales inversely with power in agreement with § 2.3, we

perform optical power measurements in the low-bias regime. We further expand this

measurement to include very high efficiency points > 1 to show that a single photon

with energy hw > kBT can be generated for less than kBT in electrical input work

as suggested by § 2.4.3. To provide further evidence that the observed optical signal

comes from heat pumping as described in § 2.2 and not emissivity modulation, we

also conduct experiments in which the LED temperature is held above and below the

detector and ambient laboratory temperatures. We then expand these measurements

83

to include the first experimental evidence of 7 > 1 operation at room temperature

and combine these results with measurements on other LEDs to support the effective

temperature concept.

The chapter is organized as follows. We begin in § 3.1 with a review of the basic

experimental techniques that will be required. These include lock-in measurements

of detector photo-current and emitter voltage, temperature control and minimizing

thermal shock, and the use of passive optical elements to improve photon collection.

Next, in § 3.2 we describe experiments that establish the basic physics of the low-bias

regime. Our goal is to show that the external quantum efficiency becomes a constant

for qV < kBT, that efficiency therefore scales inversely with optical output power,

and that this behavior continues beyond the conventional limit of unity wall-plug

efficiency. To this end, light-current-voltage (L-I-V) measurements are made on a

heated LED emitting at 2 .5 pm. Since the optical power available from the LED at

unity efficiency is much less than the blackbody background, the lock-in technique

is necessary; because the optical power also increases rapidly with emitter tempera-

ture, thermal control is required to keep the device at an elevated temperature. In

§ 3.3, we attempt to reach higher values of Lusity by increasing the emission wave-

length. Despite the wavelength-scaling of the Carnot limit we derived in § 2.4.3, we

find that increased non-radiative Auger recombination in the measured 4.7ptm LED

restricts Lunity. For this experiment, a photo-detector sensitive to longer wavelengths

is required. To maintain a low noise floor while decreasing the detector bandgap, a

smaller-area detector is used; it also incorporates a hyper-hemispherical optical im-

mersion lens that limits photo-detection to a small acceptance angle. As a result,

additional passive optics were required to achieve reasonable collection efficiency. Fi-

nally, in § 3.4, similar measurements are reported on an LED emitting at 3.4pum using

the same detector. Measurements on this intermediate-wavelength LED do show in-

creased power at unity efficiency, and permit the observation of q > 1 operation at

room temperature. We end our discussion by using data at all three wavelengths to

examine the direct connection between bias voltage V and optical power L suggested

by the effective temperature model from the previous chapter.

84

3.1 Experimental Techniques

3.1.1 Current-Biased Lock-In Technique

In order to measure the low optical power levels emitted by the LEDs in these ex-

periments, a lock-in technique was used. Such a procedure is necessary because at

sufficiently low voltages, electrically-driven light emission is smaller than the back-

ground blackbody radiation incident on the detector. Since the arrival rate of black-

body photons and the corresponding current generated in the detector are fluctuating

quantities, optical signals resulting from low forward bias voltages must be somehow

distinguished from blackbody radiation to be measured with useful accuracy. Mod-

ulating the LED allows us to separate the photo-current it produces from most of

the noise in the detector circuit. By looking specifically for a photo-current signal

with the same frequency and phase as the excitation over a long integration time,

arbitrarily small optical power signals can be measured.

In particular, in order to measure optical power from an LED in the low-bias

regime (V < kBT/q) with a signal-to-noise ratio above 1, a lock-in technique is

necessary. This is because the spectral intensity emitted from an LED is equal to

double its equilibrium blackbody intensity when a forward bias of qV = kBT - ln(2)

is applied. This may be shown in several ways, but the simplest is to calculate the

effective temperature at which the active region must glow to double the blackbody

intensity, then solve for the corresponding applied voltage:

11Cw 1 -= J(f; T*) =21I(f; T) =2 Cw 1 (3.1)exp ( ) exp -1

exp (~h*) - exp (3.2)kBT- 2 kBT)

=T ln - + h (3.3)k kBT

T* =T (I - k ( (3.4)

85

Comparing this expression with Equation 2.24, we see that the blackbody spectral

intensity at w is doubled when the forward voltage is qV = kBT ln(2). Recall now that

even with perfect optics, the best one can do is to exclude all intervals of phase space

which do not contain the signal and include only and all of the desired volume which

does contain the signal. As a result, if we take the mean of the blackbody-induced

photo-current to be noise, the preceding logic indicates that without modulating the

excitation, the best signal-to-noise ratio one can measure at qV = kT ln(2) is 1:1.

Since we want to measure optical power from LEDs at qV < kBT, we will use lock-in.

In these experiments, the LED was placed electrically in series with an unheated

resistor (5MQ, 500kQ, 50kQ, or 5kQ depending on the magnitude of current required)

and the combined load was biased with a 1013 Hz on-off voltage square wave. For the

measurements on the heated 2.1pm LED, the inverse slope of the diode's I-V curve

around the origin (i.e. it's zero-bias resistance) varied from 6kQ at low temperature

to 168Q at high temperature. Thus for the low-bias measurements at high temper-

ature, the series resistor could be chosen to dominate the load across the function

generator so that the LED was approximately current biased. The optical power was

detected by various free-space infrared photo-detectors whose photo-current signal

was amplified and measured by a trans-impedance gain stage connected to a digital

lock-in amplifier. The gain stage was composed of a commercial trans-impedance

amplifier (SRS model SR570 Low-Noise Current Preamplifier) operating in low-noise

mode with gain 2pA/V, followed by a second voltage-to-voltage amplification stage

with gain between 2 and 20. The analog filters built into these two amplifiers were

configured to form a bandpass filter with one-pole or two-pole roll-offs around 100 Hz

and 10 kHz, so that power outside this band did not cause output or input overloads

at any stage. Within the digital lock-in amplifier (Perkin Elmer model 7280 Wide

Bandwidth DSP Lock-In Amplifier), only the notch filters at 60 Hz and its harmonics

were used, and the analog gain stage before the ADC was not used. The values for

the optical power L reported here are related to the raw voltage Vr read out from

86

the digital lock-in amplifier by the following equation:

1 1 7rL = - Vraw , (3.5)

Rphoto-diode GTIA V

where Rphoto-diode is the detector responsivity in A/W, GTIA is the total trans-impedance

gain of the amplifier(s) in V/A, and the dimensionless factor of 7r/V1 is necessary

because Vraw indicates the root mean square of the first harmonic (1003 Hz) con-

tributing to the square wave whereas L refers to the height of the square wave whose

low-end value is zero.

X 10- Low-Power Light Measurements

2. - -E I I sW

a 2.5 votaerobgE SmSample 12--

>- -. .-2. -5 - ..*. .. .15.

Raw X (In-Phase Component) x 13-

Figure 3-1: Raw X and Y quadrature components of the observed photo-current gen-erated in the detector when a small forward bias is applied to the LED at 135C. The

data is not scaled to account for the detector's responsivity or intermediate amplifica-

tion; instead the data here is the raw output from the digital lock-in amplifier. Note

that data recorded when the LED was not driven indicate that the background noise

had no preferential phase relationship to the excitation signal. For all measurementsin this particular figure, a time constant of T=10s was used; the raw measurements

under a given excitation condition were recorded at intervals of At=l0s. The dataset labeled 'EMI Test' refers to measurements taken when the current source was

disconnected from the LED; its proximity to the origin confirms that the recorded

current was not the result of electromagnetic interference between the source loop

and the detector loop.

As shown in Figure 3-1, the noise was observed to be zero-mean, with no pref-

erential phase relationship to the excitation signal. The figure includes several raw

data points from the lowest-voltage lock-in measurements of optical power at 1350 C.

87

Although the blackbody background in DC measurements was substantial, the spec-

tral power of the noise around the lock-in frequency was observed to be independent

of the source temperature. Instead, the temperature of the detector element corre-

lated with the noise power, suggesting that the dominant noise source could be the

current-noise of thermal generation processes in the photo-diode. Another explana-

tion could be the dependence of the detector's shunt resistance on its temperature.

Closer consideration suggests that these two explanations in fact reflect the same

physics, as the former refers to the microscopic processes that give rise to the latter

macroscopic phenomenon. This subject is explored in greater detail in § 5.4, where a

more complete analysis of the detector noise is presented.

For each experiment, overlapping power measurements were made at higher op-

tical power using a simplified version of the setup. The AC current source (which

includes the series resistor) was replaced by a DC voltage source and the lock-in am-

plifier was replaced with a digital multimeter. In general, the lock-in optical power

measurements were in agreement with the DC to within the experimental uncertainty.

Some variation between data sets was expected due to imperfect feedback control of

the LED temperature combined with the extreme sensitivity of various measurements

to this parameter. Data acquired by both methods in the overlapping power range

appears in the figures in later sections.

3.1.2 Temperature Control

In order to measure the efficiency of photon generation by these devices as a func-

tion of elevated lattice temperature, a temperature control circuit was constructed.

The commercial LED21Sr, LED34Sr, and LED47Sr devices were manufactured in

threaded M5xO.5 metal cans (roughly a 5mm long cylinder, 5mm in diameter). For

each measurement, the can was placed inside of a copper cylinder with a recess in one

end for the LED and a recess in the other for a cartridge heater capable of heating at

around 1kW. The LED recess was not close-fit and tapped due to thermal expansion

issues, so thermal paste was necessary to ensure reasonably high thermal conductance

between this copper housing and the outer can.

88

Figure 3-2: Experimental setup for thermal feedback control of 2.1pm LED during

efficiency measurements.

89

Power to the cartridge heater was drawn from standard 120V 60Hz wall-power,

with a duty cycle set by a high-power FET whose gate voltage was controlled by a

digital P-I-D temperature controller. Input to the P-I-D controller originally came

from a single thermistor placed near the LED end of the copper housing. However,

this led to long thermal time constants and ringing of device temperature on long

timescales. To address this issue, the single thermistor was replaced by two thermis-

tors as shown in Figure 3-2. The first thermistor was placed near the heater for tight

feedback control; the second was placed near the LED for more accurate temperature

measurement.

The measured device temperature was highly uncertain. In spite of the use of ther-

mal paste, variations due to pressure applied to holding the thermistor near against

the copper surface led to differences in temperature measurement of up to 10'C at

the 125 to 135'C range. During the acquisition of the data presented in § 3.2, the

thermal impedance between the copper cylinder and the thermistor was relatively

high and the measured temperature was 125'C. Subsequent experimentation using

a metal clamp to hold the thermistor in place suggested that the actual tempera-

ture of the copper housing during this experiment was 135'C, so the experimental

temperature was reported as such. Regardless, the finite thermal resistance between

the copper housing and the lattice temperature of the semiconductor p-n junction

remains a substantial systematic uncertainty of order 10 0 C. This was deemed accept-

able because the central quantities in the high-temperature experiments, namely the

input and output power of the LED, were measured independently of this parameter.

3.1.3 Thermal Shock of LED Packaging

A major initial obstacle to repeatability of these experiments was thermal shock to

the LED packaging. Several early attempts to replicate the phenomenon resulted

in the discrete, irreversible changes to the electrical response indicative of new shunt

resistances, with the device eventually becoming an electrical short-circuit at elevated

temperatures.

Three working hypotheses were formulated, one of which was ruled out. The first

90

430 qm7 1

()Dagof LEout

Fig. F. S-egric : es s t p ot g fpr-chap diodt f k.edor sthein creier i A t hen-maSb adstvatey(2)isahet-buGed.sSb aiet. (3) ti jn p-hiCeSb wyi.(4 is reu-aSb aye (r s is e Sdid carrier a(6)s the rsion ontact

f ai ir ate UteAppcdt cabeed ' ''mictn ande (d r nd (9)F a3-3t.coclactpads wis the deposited Sio +Pb coating.

(c) Diagram of LED mount.(a) Packaged LED (front). (b) Packaged LED (back). (Taken from [10])

Figure 3-3: Images depicting possible locations of device failures due to thermal shock.Figure 3-3a and Figure 3-3b show the outer LED packaging. From the backside ofthe device, an epoxy filler is seen surrounding the leads for strain relief. A thermally-induced strain field near the junction between the wire leads and the silicon carrierwafer may be responsible for the observed device failures. Alternatively, thermalexpansion differences between the die and carrier wafer may have resulted in contactfailures at the points labeled '6 or '9' in the diagram in Figure 3-3c.

91

hypothesis was that the contact metal at high temperatures was diffusing into the

semiconductor material. This seemed feasible because the device failures were taking

place above the maximum permitted operating and storage temperatures presented on

the datasheet and the softer materials used to make long-wavelength opto-electronics

often have lower threshold temperatures for the diffusion of metals. However, because

device failures happened at various temperatures, this seemed unlikely. After raising

the matter with the growers of the device, a research group led by Prof. Matveev

at the loffe Physico-Technical Institute in St. Petersburg, we were informed that the

typical temperature for metal diffusion in the quaternary found in the cap layers of the

LED21Sr was 180 to 185'C. This was commensurate with the 43% reduction in output

power seen when the copper housing temperature was raised from 190'C to 195'C

during a measurement with a Fourier Transform Infra-Red (FTIR) spectrometer.

Because most early device failures happened at temperatures far below this, typically

between 80 and 135'C, alternative hypotheses were developed.

A second working hypothesis was the failure of solder junctions between the de-

posited contact layers and the Silicon carrier wafer shown in Figure 3-3. Attempts

were made to examine these bonds directly by machining open the M5xO.5 can of a

failed device, but the tools used were not sufficiently precise and the carrier wafer

with device was lost as scrap.

The third hypothesis remains the most likely. The epoxy designed for strain

relief of the lead wires entering the backside of the device (see Figure 3-3) could

experience an internal strain field due to the elevated temperatures and cause shunt

paths between bare wire leads buried within it to become significant.

Since very slow increases of temperature still resulted in stable light emission

at temperatures much higher than typical failure temperatures, the observed failure

mode was more characterized by thermal shock than harsh steady-state thermal con-

ditions. This is commensurate with the third hypothesis as polymer materials can

experience internal strain fields as they are heated, but slowly relax on the timescale

of minutes or hours. Temperature slew rates of less than 5K per hour were suffi-

cient to avoid this effect altogether. By incorporating these limits on slew rates into

92

the experimental protocol, the result was eventually highly reproducible, with the

remaining variations small enough to be explained by differences due to variations in

growth and fabrication processes which were present before any thermal cycling.

3.1.4 Optical Design

We have also used passive optical elements to increase the photon collection efficiency

of optical power measurements made with detectors of various sizes. For measure-

ments of optical power at A < 2.6pm, an InGaAs photo-diode from Hamamatsu

with a relatively large 3mm-diameter active area was used. At longer wavelengths,

immersion-lens photo-diodes manufactured by Vigo System were used. The effective

area of the Vigo detectors were at most 1mm2 . Because of the die size, working

distance, and divergent emission cone of the LEDs, only a fraction of the emitted

photons could be collected even by placing these Vigo detectors directly up against

the packaged LEDs.

To improve collection efficiency, a pair of lenses was used. The first, a 2"-diameter

Germanium lens with f =50mm, was placed 2f from the source and 2f from the

detector. Because the acceptance cone of the photo-detectors were > 300, further

reduction of the spot size could be achieved using a second lens. A smaller 12mm-

diameter CaF 2 lens was placed near the detector-side beam waist for this purpose.

For experiments using the Vigo PVI-3TE-6 detector with 1mm 2 active area, the lenses

were observed to improve collection efficiency by roughly a factor of 4.

Once this optical engineering had been done, it was expected that switching de-

tectors to the Vigo PVI-3TE-4 with (0.25mm) 2 would result in 10x lower noise-

equivalent power because of the shorter cutoff wavelength (4[pm instead of 6pm).

However, the large reduction in the signal magnitude due to the reduced detector

area dominated (i.e. the signal reduction was more than a factor of 10), even with

the use of passive optics, suggesting that the final spot size achieved is far from ideal

and further optical engineering should improve results.

One more set of experiments was designed to quantify the collection efficiency of

the detectors. A lensless planar 2mm x 2mm photo-conductor was used to map the in-

93

tensity field. The goal was to acquire data on the intensity as a function of transverse

position, which could then be de-convolved by the 2 x 2mm area aperture function

to reconstruct the beam profile. Initial results showed that the photo-detector was

collecting roughly 1/8 of the light from the 3.4pm LED and 1/4 of the light from

the 4.7pm LED. These measurements, however, relied on accurate knowledge of the

photo-conductor's response spectrum, which was highly uncertain near 4.7pm due

to its proximity to the detector's red cutoff. An attempt was made to use a pin-

hole to compare photo-conductor measurements against photo-diode measurements

for an intensity profile which both should collect nearly ideally. This experiment

was unsuccessful because the total optical power through the pinhole was very small

and revealed a significant noise source due to electromagnetic interference and/or a

ground loop connecting the function generator on the source side with the lock-in

amplifier on the detector side. These observations of phase-locked noise casted doubt

on all measurements made with the photo-conductor, but since no such observations

occurred with the photo-diode, this set of experiments was temporarily abandoned.

3.2 Demonstration of r7 > 1: A = 2.5pm

The first experiment we report here was the first demonstration of an LED operat-

ing above unity efficiency. As mentioned in § 1.3, the operating regime in which this

phenomenon was observed differed substantially from that of previous work. This dif-

ference is captured concisely by three characteristic energies: the electrical energy qV,

the thermal energy kBT, and the bandgap energy of the semiconductor from which

the photons are emitted Egap. The phenomenon was observed by applying a very

small forward bias of 70pV, so that qV was several hundred times smaller than kBT.

In the low-bias regime, V < kBT/q, the experimentally-measured external quantum

efficiency nEQE oc L/I was observed to become voltage-independent and further re-

ductions in voltage increased the wall-plug efficiency q = L/(IV). Previously, the

low-bias regime had been dismissed [31, 14] as producing impractically little power.

However, by moving to narrow bandgap materials and raising the ambient temper-

94

ature as Berdahl originally suggested[43] in 1985, the power available in this regime

was increased by several orders of magnitude.

This series of experiments was performed on an existing commercial device, the

LED21Sr made by loffeLED, Ltd. As described in § 3.1.1, a current-biased lock-in

technique was employed to source a small current square wave into the LED. An

uncooled Hamamatsu G5853-23 long-wavelength (A < 2.6pm) InGaAs p-i-n photo-

diode (Rpeak=1.3 A/W) with a circular active area 3mm in diameter was placed a few

millimeters away from the LED's emitting surface so that most emitted photons were

captured by the detector. Lock-in measurements of the photo-current signal and the

voltage across the LED were performed. As shown in Figure 3-4, the LED voltage

measurements were found to be in agreement with DC measurements of the zero-bias

resistance. As shown in Figure 3-1, optical power measurements at low power were

zero to within uncertainty when the source current was off, and increased linearly

with source current in the low-bias regime as expected.

Measurements of voltage and optical power were performed at various tempera-

tures using the thermal control scheme described in § 3.1.2. Power measurements

at higher current were in agreement with DC measurements to fair accuracy; both

AC and DC power measurements appear side-by-side in Figure 3-5. The uncertainty

in the optical power measurements at DC was quantified by measuring the fluctuat-

ing photo-current with the LED off but the source stage at temperature. However,

because the optical power at fixed voltage is in theory very sensitive to emitter tem-

perature, and imperfect thermal control resulted in ringing of as much as 5YC during

the measurements, the data should be expected to contain significant fluctuations of

the optical power not captured by zero-signal measurements in this case. This source

of error does not reduce the accuracy of the results, however because variations among

distinct measurements at different excitation levels would cause this uncertainty to

be reflected in the position of the data points in the final measurements as shown.

Furthermore, the uncertainty in power due to this effect would be much less impor-

tant when measurements are taken at fixed current, as in the AC measurements at

low power. We note also that because our lock-in setup was not able to source more

95

10-5

106

-a 1U

P A/ -

/CI

40

Ae 0

84'C / 25'

1350C kT/q @ 135"C

k,T/q @ 84-C

kBT/q @ 25C

10~4 10-3 10-2

Voltage (V)101

Figure 3-4: The lock-in voltage measurements in the low-bias regime were in agree-ment with the values for zero-bias resistance extracted through DC measurements.

The discrete markers indicate pairs of voltage and current that were measured bylock-in. The lines indicate the zero-bias I-V curve given by DC measurements. Thelines are dashed above 10mV, where significant deviations from linearity are expected.

96

a)L..

L..

0

0-4

-

than 2mA and we could include only DC optical measurements above 100nW and

still have meaningful data (i.e. signal-to-noise ratio > 10), measurements using the

AC and DC techniques existed over a finite range of input and output power levels.

a b

10-2Temp. Model Exper. / A o 100% Wall-Plu Efficiency25*C ---- o 1084*C -------- A Temp. Model Exper.

3 10 135*C 250C4) 10- 10- 84*C ..----- A

4)135'C -- - o

us10 -2 ,

010-10

10 -104 10-2 10 10 10~5 l-

Current (A) Light Power (W)

Figure 3-5: Efficiency measurements of the LED21Sr infrared LED at various tem-peratures. Sub-figure (a) shows the external quantum efficiency as a function ofelectrical current. Sub-figure (b) shows the wall-plug efficiency as a function of de-tected optical power. In each case, the lines denote the results of a numerical modeland the discrete markers denote experimental data. Where error bars are not visible,the measured uncertainties are too small to show.

When a 2.1V square wave was sourced across the LED at 135'C, a 0.41PA current

square wave was driven through the LED. During the on-phase of the square wave,

the forward bias voltage across the LED was just 72.5±4pV. During this phase, the

emission of 69±11pW of optical power was detected by the aforementioned lock-in

photo-detection technique with time constant r=10s. As seen in the top left of sub-

figure (b) in Figure 3-5, since just 29.9±0.lpW of was used to drive the LED source,

the wall-plug efficiency of the device at this operating point was q =2.31±0.37, and

constituted experimental confirmation of / > 1. This single measurement represents

the high-temperature low-power endpoint of a larger data set characterizing the sup-

plied current and voltage along with the resulting optical output power as the LED's

97

temperature was varied between 25'C and 135'C.

Since only the detected photons were considered as output power, corrections due

to imperfect collection could only further raise this efficiency. Furthermore, as seen

in Figure 3-6, the LED's emission spectrum gradually red-shifts out of the responsive

band of the photo-diode at high temperatures. The optical power measurements were

calculated using the detector's peak responsivity of Rphoto-diode = 1.3 A/W so that

reported optical power figures again serve as a lower bound.

1

0j0-COa)CE

C

cc

0.8

0.6

0.4

0.2

01500 30002000 2500

Wavelength (nm)

Figure 3-6: Relative intensity spectra of the LED21Sr device at various tempera-

tures. Also shown is a piecewise-linear approximation to the relative responsivity

spectrum as presented in the photo-diode's data-sheet. The peak responsivity of the

Hamamatsu G5853-23 long-wavelength InGaAs p-i-n photo-diode, which exists from

approximately 1900 to 2400nm, is 1.3 A/W. This value was used to compute optical

power from raw lock-in measurements at all temperatures because the spectra could

not be easily acquired simultaneously.

By examining the dimensionless quantum efficiency as a function of voltage across

these measurements as presented in Figure 3-7, we can also confirm our prediction

98

//

DetectorResponsivity

25 0C 190*C

that 77EQE becomes voltage-independent below kBT/q.

kT/q @ 135'C

k,T/q @ q84"OC- 1350C

840C kT/q @ 25-C

-.~EE.

10- 0-210Voltage (V)

b

/0./0

00 25'C-U -

10~1

Figure 3-7: External quantum efficiency versus LED voltage for the LED21Sr atvarious temperatures. The quantum efficiency of the LED in this experiment wasobserved to be voltage-independent for voltages less than V = kBT/q as expectedfrom § 2.3.

3.3 High Power Attempt: A = 4.7pm

While the preceding electrical and optical power measurements on the emission of

2.5pm photons demonstrated that high efficiency was possible in spite of significant

irreversibility (i.e. low 'qEQE), the application space for the phenomenon is strongly

limited by the lack of power available in this regime. As we showed in § 2.4.3, this is to

some extent a fundamental trade-off: for a given wavelength and quantum efficiency,

lower intensity light requires less voltage and allows a larger fraction of the electron-

pumping energy to come from the surrounding lattice vibrations and thereby permits

99

1 04

C4

CwE

10~3

104

10510~4 100

-

a

higher efficiency. This observation suggests that concerted efforts to improve the

quantum efficiency and moving to longer wavelengths could lead to higher power.

Because the former requires substantially more effort, the latter was the explored

first.

As seen in Figure 2-12, the Carnot limit for longer wavelength LEDs permits

higher power densities in the low-bias regime. However, as with the 2.5pm emitter,

substantial deviations from Carnot-efficient operation were observed in initial tests.

The results of low-bias efficiency measurements at various temperatures appear in

Figure 3-8.

0 -4 __ _ _ _ _ _ _ _ _ _ _010

01_0

U -5C 10 10 0

Curn)() 0pt-4ialPwe WE

w

-6

10 -310 -210- 10908Current (A) Output Optical Power (W)

Figure 3-8: Initial efficiency measurements on 4.7pm LED. At left is a plot of quantum

efficiency (?EQE) versus current (I) and at right is a plot of efficiency (,q) versus output

power (L) as a function of temperature for a 4.7pm LED. The dashed lines are best

fit curves that follow the expected q oc 1/L power law.SWdeo Veaim of AMLAB

The lack of monotonic temperature-dependence in these observations suggests

that irreversible changes may have taken place during the measurements. Sentaurus-

based transport simulations have suggested that temperature-dependences may in

fact change sign, but those results did not qualitatively fit the observed data either.

In those simulations, SRH recombination was the primary non-radiative pathway at

low temperature, and at high temperature Auger became the primary non-radiative

pathway. As a result, a maximum of quantum efficiency was seen with respect to

temperature rather than a minimum, suggesting an alternative explanation is required

to explain the observations here.

After these measurements were taken, tests on other long-wavelength devices re-

100

vealed that high currents may result in irreversible damage and the measurement

protocol was changed to avoid this. At the time of the collection of this data, how-

ever, this failure mode was unknown. If such an irreversible change was responsible

for the reduction in efficiency between the 30'C and 60'C measurements, the re-

producible aspect of the temperature-dependence of device performance would be

indicated by the differences between the 60'C and 100'C data. In this case, the

observations suggest that temperature is indeed still increasing both EQE and 77.

Although our investigations into these devices remains incomplete, the low ob-

served quantum efficiency suggests that shorter-wavelength sources may in fact gen-

erate higher intensity light at unity efficiency. Observations to date suggest this is

due to improved material quality and the improved low-bias quantum efficiency that

results from it.

3.4 Lower Emitter Temperatures: A = 3.4ptm

Various experiments were performed on another commercial device (LED34Sr), this

time at lower emitter temperatures. Some experiments were designed to refute al-

ternative explanations for the observations at 2.1pim, while others were intended to

confirm that the phenomenon was observable with the emitter at room temperature.

The work described here formed the basis for a journal article [83] published in 2013,

and the structure of our discussion mirrors that of the article.

3.4.1 Exclusion of Emissivity Modulation

One of the chief criticisms of the experimental technique at 2.5pm was the possibility

of the detected signal originating in a modulation of emissivity for blackbody radiation

rather than thermo-electric pumping of the device active region. By attempting

to observe the same phenomenon in a configuration in which the emitter was not

much hotter than the detector or the other surfaces surrounding the experiment,

we attempted to exclude this alternative explanation. Since lower temperatures are

required to thermally generate carriers in a smaller bandgap material, but much

101

smaller bandgap materials appeared to have too high of defect densities to permit

sufficient ?7EQE, an LED emitting at an intermediate wavelength of 3.4pm was chosen

for this task.

In Figure 3-5, the q oc 1/L scaling is observed over 3 orders of magnitude in

output power (6 orders in input power) at 135'C, extending all the way down to

the Noise-Equivalent Power (NEP) limit of the photo-detector circuit. While this

does combine with the modeling results presented alongside the data and the theory

offered in Chapter 2 to present strong evidence for the scaling law to continue to

arbitrarily low power, any physical effect that might result in a phase-locked photo-

current whose magnitude is linear in the excitation current could in principle create

this effect. In particular, the possibility has been raised that the signal may result

from a small modulation of the emitter's emissivity with current. As mentioned

previously in § 3.1.1, the observed signal at low bias is necessarily much smaller

than the blackbody radiation power flowing out from the LED surface and onto

the detection surface. When V = 70p-V, this ratio is just a few parts in a thousand,

meaning that even a (spectrally flat) 1% modulation of surface emissivity with current

would be more than sufficient to explain the observation. Nevertheless, modeling and

theory suggest this is not the explanation, as do the following results.

Precisely what is meant by "emissivity" in this context is not entirely clear because

the term is a macroscopic property while our description to this point has been

primarily microscopic. For the purposes of this discussion, we regard a microscopic

model as one which refers to discrete particles and which describes a physical state in

reference to the quantum state of the complete many-body system, even if that state

is not a pure state. When discussing a macroscopic quantity like emissivity alongside

such microscopic models, we must be careful to define the terminology explicitly.

We use the term "emissivity" to refer to the degree of energetic coupling between

a body at thermal equilibrium and outgoing radiation modes. A body's emissivity,

when combined with a body's temperature, determines the power density of thermal

radiation it emits, and unless otherwise stated the presumption is that the spectral

intensity of the emitted radiation is proportional to a blackbody radiator of the same

102

temperature (i.e. the emissivity is a constant with respect to wavelength).

The emissivity c is defined with respect to radiating body which is at thermal

equilibrium at some temperature T, and so we must specify how we will generalize

these concepts to a non-equilibrium body such as an LED under nonzero applied

voltage. The emissivity is conventionally a constant independent of the excitation

of the system while the temperature contains all the information about its level of

thermal excitation. For this reason, we choose to generalize the concepts by using

the emissivity to refer to any change which is not an excitation of the body itself.

For example, applying a voltage which brings the electron-hole subsystem out of

equilibrium with the lattice would not refer to an emissivity change, but a change in

the surface reflectivity would. Using this definition, we now examine the compatibility

of the alternative interpretation of emissivity modulation with experimental results.

We organize the changes of the LED state corresponding to an emissivity change

into two categories. Neither a perfectly transparent body nor a body with a perfectly

reflective surface permit energy to flow out of the internal degrees of freedom of a

finite-temperature body into radiation modes in free space. Thus either a modulation

of the LED's transparency or its surface reflectivity at the wavelengths at which the

photo-detector is sensitive would constitute a modulation of emissivity.

First we consider a transmission modulation. Since the LED in the high-temperature

experiment was housed in opaque packaging, which was in turn held in a recess within

a heated copper rod, all of the bodies behind the device whose emission would be seen

in place of the LED's when the LED's transmission was increased were at the same

temperature as the device. The experimental procedure could not ensure that the

temperature was exactly the same, but if the temperatures were equal, there would be

no signal at the photo-detector. If the housing seen through the device were taken to

be slightly higher than the LED, such an effect could produce a photo-current of the

sign that was seen. However, the temperature of the setup was always ringing around

the set-point temperature under the control of the Wavelength Electronics LFI-3751

P-I-D feedback controller, so the magnitude of this signal (increasing with the temper-

ature difference between LED and its housing) would vary in sync with this ringing.

103

If the ringing brought the housing to a temperature below that of the active region,

that would even cause a sign change. Since no behavior of this type was observed

in the high-temperature experiment, the transmission modulation interpretation is

inconsistent with experiments.

The reflectivity modulation possibility must be considered in each of two sub-

cases: those in which the device's surface reflectivity is primarily specular and those

in which it is primarily diffusive. These two possibilities are addressed by the following

experiments.

Of these we first consider specular reflection. In this case the temperature of

the absorptive detector surface would affect the photon flux returning to it from the

emitter and thereby affect the measured photo-current. In this case, the rays which

would depart the surface of the device on a trajectory which eventually lands on the

absorptive photo-responsive surface of the detector could originate in modes of iden-

tical transverse position at the emitter surface, identical transverse momentum, and

longitudinal momentum of the opposite sign. Since the emitter and detector struc-

tures, as well as the intermediate optics, possess symmetry under inversion in the

transverse dimensions (i.e. they are all circles or squares), under perfect alignment

these light rays would all originate at the detector surface itself. Thus the tempera-

ture of the detector would be relevant to the signal seen if this reflectivity were being

modulated by our current source. If the detector temperature were colder than the

emitter temperature, then replacing photon flux from the LED with reflected pho-

ton flux from the detector would lead to a decrease in measured photon flux with

increasing reflectivity. Since the lock-in measurement of the photo-current indicated

an increase in photon flux with voltage, the sign of the reflectivity's dependence on

applied voltage would need to be negative (i.e. MRsurf/V < 0). Likewise, if the

detector temperature exceeded the LED temperature, the phase of the photo-current

signal should flip sign.

This situation was tested with the 3. 4 pm LED at room temperature and the TE-

controlled photo-diode held above and below room temperature. In Figure 3-9, raw

data from two nearly-identical experiments are shown side-by-side. The displacement

104

-10X 10< 4

470C PDOFF ON01....... 4 .

-0 0O a

.L -L *CP

00-c0

(/3-4

2 0 2 4 6 8In-Phase Photo-Current (A) x 1o o

Figure 3-9: In-phase and out-of-phase components of the photo-current signal for twosimilar measurements, one with the detector hotter than the emitter and one with itcolder. As the detector temperature was varied from above the emitter temperatureto below, the phase of the optical signal did not undergo a 1800 shift, indicating thatthe observed signal is not due to a modulation of a specular surface reflectivity. Themeasurement was taken with an excitation voltage of 4.4mV and a current of 2pA,placing it clearly in the low-bias regime. The solid markers denote measurementsof the amplified photo-current signal with the LED off; the open markers denotemeasurements of the same signal with the LED on. The the phase of the optical powersignal is near zero for both detector temperatures. The magnitude difference resultsfrom the temperature dependence of the detector's responsivity, which we explore ingreater detail in § 5.4. As explained in the text, this is compatible with an electro-luminescent cooling signal but not with a specular surface-reflectivity modulationsignal under good optical alignment.

105

of the solid blue squares from the open blue squares near the origin indicates the

presence of a phase-locked photo-current signal when the detector temperature was

well below ambient (-50 C). The displacement of the solid red circles from the open

red circles near the origin indicates the detection of a phase-locked photo-current

signal when the detector temperature was well above ambient (+470 C). Clearly the

sign of these two signals is the same. Since the phase-locked photo-current signal did

not in fact flip sign, we conclude that the preceding explanation is incompatible with

the observations in Figure 3-9.

Copper

Thermocouple /Housing LED

TEC--

Lens Uncooled (297K)Photo-detector

U.

LLZ

294KIED

E 0 297K LED

0* 3WOKLED -16 -8 0 8 16In-Phase Photo-Current (pA)

Figure 3-10: At top: a diagram depicting the experimental setup for the experiment

in which the temperature of the LED was heated above and cooled below ambient.

At bottom: the in-phase and out-of-phase components of the resulting photo-current

signal. The meaning of the different markers is stated explicitly in the legend to the

left of the plot and follows the same conventions as Figure 3-9. Results indicate that

the sign of the photo-current did not change with the sign of the LED-to-ambient

temperature difference, refuting the interpretation of the electro-luminescent cooling

measurements presented throughout this chapter as instead originating in a voltage-

controlled modulation of the surface reflectivity of the LED.

106

-....................... ..............

* ®r

.. ....... ......

24

Next we consider the case of diffusive reflection, which also covers the case of spec-

ular reflection under poor alignment. In the case of diffusive reflection, the rays which

would land on the absorptive detector surface could originate from ray incident on the

surface. Since the environment is roughly in equilibrium at 297K (and the blackbody

flux averaged over the detector's photo-responsive band was not significantly higher

than this in other observations), in analogy with the previous test, we performed sim-

ilar measurements in which the sign of the temperature-difference of relevance was

changed. In this case, because a changing of the diffusive reflection coefficient would

affect the fraction of the emerging photon flux which originated in the device or the

surrounding environment, we chose to raise and lower the temperature of the LED.

As shown in the plot at the bottom of Figure 3-10, we see that once again the sign

of the photo-current signal did not flip with the sign of (TLED - Tambient). From this

observation, we infer that the measurements reported throughout this chapter are

not compatible with originating in the modulation of the LEDs' diffusive surface re-

flectivity or specular surface reflectivity under conditions of poor alignment. In fact,

since the detector in these measurements was not cooled, the data in Figure 3-10

alone stands in contradiction to the effect of any type of surface reflectivity regardless

of alignment.

We now pause to clarify the role of the preceding arguments concerning emissiv-

ity modulation within the broader experimental effort described in this chapter. We

regard the preceding arguments as reasonably strong evidence against interpreting

our measurements as emissivity modulation and supporting evidence for the obser-

vation of electro-luminescent cooling. It is not easy to entirely exclude any family of

interpretations with a few data sets like the ones found in Refs. [47] and [83], let alone

prove a single interpretation beyond doubt. The consistency of the data with predic-

tions from theoretical models, combined with the reproducibility of the measurement

using LEDs in various material systems, with different detectors, and at different

temperatures is also a significant contribution to our confidence in the interpreta-

tion. More specifically, the theoretical models quite clearly predict that the quantum

efficiency of an LED should become independent of voltage for qV < kBT. The

107

value for qEQE indicated by the data was approximately 3.30x 10-4 and 3.35x 104

for the 2.15pm LED at 135'C and the 3.4ptm LED at room temperature respectively

(statistical uncertainty for both ?7EQE values was between 1 and 2 x 10-). In both of

these experiments, literature data suggests that SRH recombination is the dominant

recombination process by orders of magnitude at low-bias. As a result the external

quantum efficiency depends primarily on the SRH lifetime r, the bimolecular recom-

bination coefficient B, and the efficiency of photon collection. Since literature values

of the first two quantities and reported values for the third are commensurate with the

observed 7EQE, we see it as unlikely that not only would this calculation be inaccurate,

but that some unspecified physical mechanism would lead to measurements with not

only the correct 7 oc scaling over 3+ orders of magnitude, but with the a constant

factor very nearly equal to what one would predict for the electro-luminescent cooling

effect value using figures from literature. Further independent measurements of the

effect, such as spectral shift due to band gap narrowing or lattice cooling of surround-

ing matter, would provide further evidence. We take up this and related topics in

Chapter 6.

3.4.2 Unity Efficiency at Room Temperature

This section presents the results of room temperature measurements of mid-infrared

LEDs at low bias voltages. We find that the results mirror those of @ 3.2 and further

support the theory presented in Chapter 2.

We begin by describing the LEDs used in the experiment. Two devices were tested,

one emitting with a center wavelength near 3. 4 pm and the other near 4.7pm. The

devices were grown and fabricated by the research group of Professor Boris A. Matveev

at the Ioffe Physico-Technical Institute in St. Petersburg, Russia. The devices were

originally acquired through a North American distributor named Boston Electronics,

but the labeling of individual devices allowed the Ioffe group to provide details of

the device fabrication beyond those made available to commercial customers. For

completeness, we provide the original text provided by the loffe group, with minor

changes to the language to enhance readability:

108

X = 4.7pm A = 3.4um

Zn-dopod p-InAsSb: p=5e17 5 pm Zn-doped p-InAs,, Sb(,()P,,: 3.5 pm

more InAsSb P=2e17 to 5e]-7 cm

N 50-60 pm Nmminally ndoped 7 m

35OiimbP .:el t =I~

more 1nP

n-type InAs (111): Sb-doped W nAs (100):

,n=2e16cm 350 m n=3el8 to 6e18 cm 200 m

Figure 3-11: Layer stacks for the 4.7pm LED (left) and the 3.4pm LED (right). Data

in these figures is derived primarily from communications with Prof. B. A. Matveev

found in the main text.

"The 4.7pm light source (grown on wafer #236) was made from an

80 pm thick narrow gap InAsSbP/InAs hetero-structure. Thin (350pm)

n-InAs wafers (n = 2x1016 cm 3 , initial dislocation density Nd 10 4

cm-2) with (111)-oriented surfaces were used as substrates. Due to the

high Phosphorus segregation coefficient, the InP concentration diminished

within a 50-60pm-thick 'undoped' n-InAsSbP layer providing the energy

gap decrease along the growth direction with energy gap gradient VEgap

of about 1-2 meV/pm. Zn was used as a p-dopant for p-n junction forma-

tion at the final stages of the growth with the resulting distance from the

p-InAsSb surface of about 5[tm. At the hetero-junction, the layer lattice

constant a was nearly the same as for the InAs substrate (lattice mis-

match Aa/a < 0.05%) while the narrow band part of the structure (i.e.

the p-InAsSb region) was lattice mismatched with respect to the InAs

substrate. Due to high InAs plasticity at the growth temperature (650-

720 C) the p-InAsSb(Zn)/n-InAsSb-InAsSbP (p ~5 x1017 cm~3 , n ~ ' 7

cm-3) graded structure formation was accompanied by stress relaxation

via substrate bending providing an 'inverse' dislocation distribution across

the hetero-structure. That is, when the plastically deformed/bent InAs

substrate was finally incorporated with the bent graded layer of high crys-

talline quality, the epi-layer dislocation density Nd didn't exceed 105 cm~2

while Nd in the substrate was as high as >107 cm- 2 [84]."

PROF. BORIS A. MATVEEV

PRIVATE COMMUNICATION ON AUGUST 21, 2012

109

At left in Figure 3-11 is a visual representation of the device's LPE-grown (Liquid

Phase Epitaxy-grown) layer stack. After growth, the diode was packaged with an

immersion lens in the same way as the 2.15jpm LED in § 3.1.3. Wet photo-lithography

was used to etch a square 150pmx150pm mesa structure (roughly 25-30Qm deep).

A Cr-Au(Zn)-Ni-Au reflective (R=0.6) anode contact was used to reflect light back

through the substrate. To improve photon extraction and create a narrower beam

profile, a nearly hyper-hemispherical Silicon lens was attached using a chalcogenide

glue with an index m2.4.

Regarding the 3.4pm source, the following description was provided:

"The layer stack of the 3.4pm light source (#6341) was a single hetero-

junction structure consisting of a 200pm-thick heavily doped n+-InAs,(100)-oriented transparent substrate doped with Sn to n ~ (3 - 6) x 1018

cm- 3 , followed by two epitaxial layers. These two layers included a 7 Jtm-

thick n-InAs active region and a 3.5pim-thick wide-gap p-type Zn-doped

(p = (2 - 5) x 1017 cm- 3 ) InAsSbP cap layer. The alloy composition

of the cap layer was approximately 73% As, 9% Sb, and 18% P. Further

information on this growth is described in Ref. [85]."

PROF. BORIS A. MATVEEV

PRIVATE COMMUNICATION ON AUGUST 21, 2012

At right in Figure 3-11 is a visual representation of this layer stack, designed to

become an LED emitting at 3.4[tm. This growth was subsequently processed in much

the same way as the previous 4.7pm growth, except that the dimensions of the square

mesa were 230x230pm.

Both the 3.4 and 4.7pm LEDs were studied across a wide range of operating

points at room temperature. The forward bias voltage, current, and light output

were measured for each device across five orders of magnitude in current, extending

from conventional operating points where the applied bias voltage qV is on the order

of the bandgap energy Egap down to the low-bias regime. The results are shown in

Figure 3-12.

110

10--6

10oc 2 ** /A 3.4pm

0L -8 440A4g-10 4.cc

0 A Current(pA)o A k?0.05 0.1 0.15

A22.15p±m0 -12 F -

.4.4

0 10-0 8--

10 10 10~- 10~- 10~4 10-2Input Electrical Power (W)

Figure 3-12: Output optical power versus input electrical power for three room tem-

perature mid-infrared LEDs. For the device emitting at 3.4pm (area 5.29x 104 cm 2,

wafer #6341) and 4.7pm (area 2.25x10- 4 cm 2 , wafer #236), the power at unity effi-

ciency was high enough to be directly observed in our lock-in measurements. For the

device emitting at 2.15pm, it was not. Note: Data for the 2.15pim LED is from § 3.2.

Insets: (top left) Relative intensity spectra for the three devices at room temperature;

(bottom right) cooling power versus current for the 3.4pim LED at room temperature.

111

The experiments on these devices were quite similar to those previously performed

on the heated 2.15[tm LED. For current levels up to 2 mA, a 1013 Hz square wave volt-

age source was used in combination with a series resistor of magnitude greater than

either device's zero-bias resistance (sometimes called the 'shunt resistance' though we

avoid that term here because this conduction is necessarily not due to shunts alone).

Optical power was measured via lock-in zero-bias photo-detection with time constants

ranging from 500 milliseconds to 500 seconds. For higher current measurements, a

DC source-meter was used along with zero-bias DC photo-detection. DC measure-

ments of current, voltage, and optical output power were in fair agreement with AC

measurements; both types of measurements appear together in Figure 3-12.

For comparison, Figure 3-12 also includes a theoretical curve representing the

Carnot limit for an emitter with the same wavelength, active area, and temperature

(298K) as the 3.4pm LED data presented. We take the idealized emitter to be

optically thick at the emission wavelength, so that the theory in § 2.4.1 may be

used to relate optical power density to Carnot efficiency.

From § 2.4, we know that for mid-infrared LEDs with a given 77EQE, Lunity should

increase as the photon energy decreases or the lattice temperature increases. Lock-

in power measurements on the 3.4pm LED showed that Lusity increased with Tiattice

from 300 K up to around 420 K. Above 420 K Lunity decreased with Tiattice, suggesting

that the increases in power at fixed voltage were likely outweighed by decreases in

quantum efficiency from non-radiative recombination and leakage, and the increased

importance of parasitic effects from contact resistance. As shown in Figure 3-13,

this temperature dependence indicates that for the 3. 4 pm LED, Lusity is maximized

when hw/kBT is around 10. Although this does not seem to be fundamental, various

authors have argued for much smaller[43] and much larger[42, 14] values of hw/kBT

without experimental realization, so this phenomenological observation may serve as

a guide for further experiments.

112

10-9 A=2.5pmA=3.4pm

-10w 10 00

c10~

4---.A=4.7pm00.

6 9 12 15 18hw / kBT (dimensionless)

Figure 3-13: Optical power density at unity wall-plug efficiency versus the dimension-less ratio Egap/kBT. The triangles labeled 2.5ptm correspond to the high-temperatureresults from § 3.2; the circles labeled 3.4pm correspond to results from § 3.4 and

similar experiments at elevated temperatures; the square labeled 4 .7 pm corresponds

to experiments on an LED of the same model as the one characterized in § 3.3.

113

3.4.3 Does Voltage Determine Brightness?

0

0110

b 100.

10-121

10~4 10~5-3

10-3Voltage (V)

10-2 1- 100

Figure 3-14: Optical power density versus applied forward bias voltage for three mid-

infrared LEDs. The discrete markers denote experimental data for voltages up to half

the bandgap energy per electronic charge q. The solid lines correspond to numerical

calculations based on Equation 2.45 and Equation 2.24.

In Chapter 2, we presented a transport model for thermo-electrically pumped

LEDs at voltages well below the bandgap. In this model, the Fermi level separation

in the active region leads to an excess population of electrons and holes, which can

also be described by a temperature at each above-gap transition energy. This temper-

ature, T*, is the thermodynamic temperature seen by fields which interact through

inter-band transitions, so that the spectral power density of photon emission may be

directly related to this value. In Equation 2.24, we made the simplifying assumption

that AEF ~ qV (true when the junction resistance dominates over parasitic series

resistances, as in most LEDs with reasonably large bandgaps and low voltages). In

114

00

=4.7pm 00

a t A~M.4m A=2.15pm.

10~ -

this way, for an emitter with a known bandgap energy and lattice temperature, the

power density above the blackbody background should be fully determined by T*,

and therefore V. Here we seek to test this hypothesis experimentally.

We also note that in contrast with the optical power density, the current density is

not entirely determined by the bandgap energy and voltage. The presence of material

defects leads to a device-specific quantity of current flowing through trap-assisted non-

radiative recombination pathways in parallel with the known quantity of net radiative

recombination we have just described.

In Figure 3-14 we compare the results of these room temperature power mea-

surements with calculations based on Equation 2.45 and Equation 2.24. For each

LED, experimental data is shown for voltages from zero up to half the bandgap en-

ergy per electronic charge. Across this range, the data is in qualitative agreement

with numerical calculations. We note that the active area of the photo-diode used

at 3.4 and 4.7[pm was significantly smaller (1 x 1mm) than that used at 2 .15pm (3

mm outside diameter). Thus the longer-wavelength measurements may include the

effect of imperfect collection efficiency by the detector; we have not corrected for this

possibility in any of the data in Figure 3-14, but initial measurements with a lensless

photo-conductor indicated this effect may have reduced the signal by a factor of 6.7.

At higher voltages, series resistances and other rate-limiting transport processes

cause L to fall short of the calculations based on V rather than AEF, including those

based on Equation 2.24. We note that our simple model must break at some voltage,

since as qV approaches Egap and the band-edge states approach inversion, T* and L

diverge in a non-physical way. We take up this discussion briefly in Chapter 6.

115


We began this chapter by detailing various experimental techniques required to in-

vestigate the wall-plug efficiency for photon generation by mid-infrared LEDs at low

intensity. We then reported a number of experimental results related the general phe-

nomenon of thermo-electric pumping in these devices. These results included the first

known experimental confirmation of electrically-driven light emission from a diode in

excess of the electrical power used to drive it.

We began in § 3.1 by presenting a series of hurdles encountered during these

experiments and the solutions developed to address them. Some of these techniques

were fundamentally needed to execute the desired experiments, such as the lock-in

photo-detection technique explained in § 3.1.1 and high-temperature feedback control

of the emitter diode developed in § 3.1.2. However other techniques were developed

in response to unexpected hurdles, including limiting the temperature slew rate to

avoid irreversible damage to the emitting diodes from thermal shock and introducing

extra free-space optical elements to enhance the photon collection efficiency of our

smaller detectors, developed in § 3.1.3 and § 3.1.4 respectively.

We then used these these techniques in § 3.2 through § 3.4 to examine the effi-

ciency of three mid-infrared LEDs across a range of temperatures from 300 to 400

K. Although data across several orders of magnitude in current density was acquired,

the focus of our experiments was on the behavior of these devices in the low-bias

regime, where qV < kBT. The results of these experiments confirmed the primary

hypotheses from Chapter 2, that an LED behaving as a thermodynamic heat pump

would have wall-plug efficiency that increases with temperature and would emit more

optical power than the electrical power used to drive it. We saw that for the longest-

wavelength emitter, the indium arsenide antimonide 4.7ptm LED from § 3.3, the

benefits of increasing exp(-Egap/kBT) appeared to be outweighed by increases in

Auger recombination and leakage as well as the increased relevance of parasitic series

116

resistances in diodes with large saturation currents. A cursory meta-analysis sug-

gested that the maximum power at unity efficiency was found in devices whose ratio

of bandgap energy to thermal energy was approximately 10.

Despite the low power density at which the thermodynamic behavior was observed

in these light-emitting diodes, these experiments served to establish a new direction

in research on the phenomenon of electro-luminescent cooling. The space of operat-

ing points explored here, those with bias voltage much less than the thermal energy

(i.e. the low-bias regime qV < kBT), not only provides a platform for experimental

demonstration, but is where the greatest deviations from conventional q < 1 behavior

are found. We found that not only are very high efficiencies (q > 1) only possible

at low voltages, but that in the low voltage limit the efficiency is required to diverge.

While the fundamental trade-off between power and efficiency demands that very

high efficiencies be associated with correspondingly low power densities, the experi-

ments reported in this chapter suggest that thermo-electrically pumped LEDs may

be more naturally suited to applications in which efficiency is more important than

power density. In Chapter 4 we will explore one such application: using LEDs oper-

ating at very high efficiency to explore the limits of energy-efficient classical photonic

communication.

117


118

Chapter 4

Communication with a

Thermo-Photonic Heat Pump

In this chapter we explore the implications of heat-pumping behavior in LEDs for the

energy-efficiency limits of classical photonic communication. In § 4.1 we revisit data

from high-efficiency measurements from our setup in Chapter 3 to motivate the topic.

In § 4.2 we calculate the minimum amount of work required for a Carnot-efficient heat

pump to encode a bit into the electromagnetic field at finite temperature. In § 4.3,

we present an experimental demonstration of a low-biased LED communication link

in which the source consumes just a few tens of femtojoules per bit.

4.1 Power Measurements as Slow Communication

In order for the photo-detector in the power measurements in Chapter 3 to detect an

optical signal with nonzero signal-to-noise ratio (SNR), the information about whether

or not the LED under test is on must be shared across the optical path. In essence,

we may relabel the LED whose power is being measured as the transmitter and the

photo-diode with amplification and analog-to-digital conversion as the receiver, and

call the entire setup a communication link.

119

At low power, where above-unity efficiency is seen, long time constants are required

to achieve an SNR above 1:1. As a result, we may expect that the bitrate for any

communication across this link would be very low. Nevertheless, the rate of electrical

power consumption is also quite low, suggesting that the amount of electrical work

required per bit could be small enough to motivate certain practical applications.

In § 4.1.1 we begin by performing a sample calculation on an actual above-unity

efficiency power measurement. In § 4.1.2 we extrapolate these results to the low

power limit to find the minimum work required per bit of information detected by the

receiver. Finally, since the devices are still far from Carnot-efficient (i.e. r/EQE < 1),

in § 4.1.3 we modify these calculations to extrapolate results for a theoretical device

with perfect quantum efficiency r7EQE = 1-

4.1.1 Sample Calculation

For this sample calculation, we will use the second-lowest-power data point from the

150'C measurements of the LED emitting around A = 2.5[tm. Note that this is also

the data set behind Figure 1-5. The raw data for that point and the two around it

appears in Table 4.1.

Optical Power L (pW) Std Dev L, (pW) SNR Electrical Power IV (pW)

11.367 3.1838 3.5703 0.19875

36.336 3.0214 12.026 1.7028

123.38 2.5508 48.369 18.940

Table 4.1: Selected power measurement data from 150'C LED emitting around 2.5pm.These are the three lowest-power measurements from this data set. The time constantfor all three lock-in measurements was Is.

From the data in the second row, we see that an optical signal with SNR~12 can

be generated using just 1.7028 pW. We may estimate the electrical work W required

120

to send this signal by the product of the input power and the time constant; this

yields W = 1.7 pJ.

To calculate the number of bits of information transmitted, we must know the

probability of an error. That is, the probability that a sent '1' will turn into a

detected '0' or vice versa. Consider the histogram of the 36 pW data point found in

Figure 4-1. The 'off' measurements are clearly separate from the 'on' measurements,

so the number of data points available is insufficient to find an error. Nevertheless,

by fitting these histograms to Gaussians and defining a decision boundary that makes

error rates symmetric, we may estimate the probability of such errors. In this case,

the two Gaussians are separated by roughly 12 standard deviations. For a simple

on-off keying (OOK) scheme in which the '0' and '1' symbols are equally likely to

be sent, the optimal decision boundary falls halfway between the means of the two

Gaussians, or just under 6 standard deviations away from each. Thus we can calculate

the probability of seeing a '1' when a '0' is sent (or vice versa) as the integral of the

normal distribution's tail starting from 6 standard deviations out. In Table 4.2 and

the adjoined caption, we express the resulting joint probability mass function (PMF)

describing this scenario. The joint PMF P is a function of the transmitted symbol x

and the received symbol y, and contains the probabilities of all possible outcomes of

a single symbol transmission event.

From the joint PMF in Table 4.2, the amount of information shared across this

channel may be calculated as the mutual information I. In standard information

theoretic notation, I is defined as the Kullback-Leibler (K-L) divergence DKL between

Px,y(x, y) and it's product-of-marginals Qx,y(x, y) = Px(x) - Py(y). Intuitively, the

K-L divergence is like a measure of distance between two probability distributions

(although it lacks basic properties like symmetry). The K-L divergence between two

probability mass functions fA(a) and fB(b) (defined over the same set of events {i})

121

5

0-

5

00

off7

1 2Raw Lock-In Signal

Figure 4-1: Histogram of the raw 'R'-values from the 36 pW output power datapoint. The blue histogram blocks at left are from measurements with the LED off.The blocks at right are from measurements with the LED on. The two red curvesrepresent Gaussian fits to this data which we use to calculate the information contentof the signal. Twice as many off measurements were made as on measurements.However, the standard deviations of the best-fit Gaussians were similar, suggestingthat sufficient data was available for a fit.

122

1

1IV=1.7pW, L=36pW

oLn

3x 10

--------------

0

x=0 X=1(Send '0', LED off) (Send '1', LED on)

y=O Pxy (0, 0) = Pxy (1, 0) =(Receive '0', LED looks off) 0.5 - -I(-5.895) i4(-5.895)

y1Px'y (0, 1) = Px'Y (I, I) =

(Receive '1', LED looks on) !D(-5.895) 0.5 - -J'(-5.895)

Table 4.2: Joint probability mass function (PMF) for communication at 36 pW. Here1(x) denotes the cumulative distribution function of a Gaussian distribution withmean 0 and standard deviation 1. The probability mass in one tail of a Guassiandistribution starting 5.895 standard deviations from the mean is D(-5.895) ~ 1.9 x10-9.

is defined as:

DKL(fAI IB) fA(ai) log (AfB(bi) (4.1)Vi

where a2 and bi are the values of A and B for each event i.

The joint PMF P expresses the probability of every event in the sample space,

where events are defined by what is sent and received; the product-of-marginals Qexpresses the combination of two distributions formed from events defined in terms

of what is sent or received, but not both. That is to say, P describes our channel,

while Q describes a channel which looks the same from either side (transmitter or

receiver), but through which random noise prohibits any information from being

communicated. The divergence (i.e. the difference) between these two situations is

defined as the mutual information I for the channel:

I = DKL(PQ) E PX,Y (X, y) - log PXY (4.2)Xy=O,1} Qx'Y (X, y)

where the logarithms in the above expression are base 2 and I is in units of bits. The

result of applying Equation 4.2 to Table 4.2 is very nearly 1, indicating that almost

one full bit of information is conveyed with each symbol transmitted. A more exact

123

calculation follows.

Defining 6 = (D(-5.895)/2, we find that the diagonal terms contribute to I as

follows:

Px, y(0, 0)Qx'y (0, 0)

= (0.5 - 6) - log

=(0.55-6)= (0.5 -6) -log(2) + log0.-6)L (0.5 ]= (0.5 - 6) [log(2) + log (1 - 26)).

Using the common expansion of base-2 logarithms log(1 + x) = 1 X + (x 2),

= 0.5 - 6 + 0.5 log(1 - 26) - 6 log(1 - 26)

1= 0.5 - 6 + 0.5 -n(2) . (-26) +...

= 0.5 - I + n 2) 6+ O(62).In(2)/

The contributions of the off-diagonal terms may similarly be ordered in powers of 6:

Px y(0, 1) - log 'Qx'y (0, 1)

=6 - log (.2)

6 log(6) + 26.

I = 1 - 26log(1/6) S1) 6 + 0(62). (4.3)

Using this expansion for the 36 pW measurement, the information contained in a

power measurement is roughly 1 - 5 x 10-9 bits. Direct numerical evaluation yields a

124

Px y(0, 0) - log0.5 - 6

0.25

Thus,1

-2 n (2)

similar result. Thus the amount of work required to communicate a bit, W/I ~ 1.7

pJ or about 3 x 108 times the thermal energy kBT.

4.1.2 Extrapolation to Low Power

As we saw in the derivation of the Landauer limit (kBTln(2) per bit) in § 1.4, the

most energy-efficient communication happens at low power.

This fact can be intuitively expected from considering communication with an

irreversible (i.e. not heat-pumping) transmitter over a time slice sufficient to send

just one symbol. As the signal power P becomes much larger than the noise power N,

the amount of energy consumed in this time slice scales linearly with P. Meanwhile,

the number of distinguishable quantization levels at a given bit error rate scales

linearly with P, so the number of bits of information scales as log(P). Thus we

expect communication protocols with P >> N (i.e. high SNR) will be further from

the minimum energy consumption per bit of mutual information shared across the

channel under consideration.

With that in mind, let us re-examine the result from the previous section. Consider

an experiment performed in the same configuration as the 36 pW data point discussed

above, but with much lower drive current through the LED. Since the output power

and photo-current are linearly related to the drive current, they would also be much

lower. However, the noise-equivalent power of the receiver is set by thermal processes

within the photo-diode, so for a fixed amplifier configuration and lock-in time constant

the uncertainty in power measurement should be fixed. As a result, the SNR should

decrease along with the input power.

To find the minimum electrical energy per bit for this link then, we must perform

the following calculations:

* For an arbitrary SNR r, find the amount of electrical work W consumed over

the time constant for the theoretical power measurement At.

125

" For an arbitrary SNR r, find the mutual information I shared across the channel.

" Find limr o . (Note: This amounts to a communication protocol in which

the signal-to-noise ratio is much less than 1, so much less than one bit is com-

municated with each symbol. Since such a protocol would need to be repeated

to communicate any practical amount of information, it may have too low of

a bitrate for practical systems. Ve consider it here primarily for its scientific

value.)

Recalling that in the low-bias regime n - 1/L, we may write a simple expression

for W in terms of the output power at unity efficiency Lunity:

L L2W = - . At = . At .(4.4)

77 Lunity

In terms of the signal-to-noise ratio r = L/L, (where L, is the standard deviation in

the light power measurements, as it was in Table 4.1), then we have:

r 2 L2W = " - At . (4.5)

Lunity

For the mutual information calculation, consider an analogous version of Figure 4-1

for arbitrary r. If we consider the 'on' and 'off' states to see the same uncertainty

L, as before (reasonable because the noise physically originates from an additive pro-

cess), we may again place the symmetric decoding boundary halfway between the two

means. As a result, the off-diagonal elements of the corresponding joint probability

distribution will be given by the probability mass in the tails a Gaussian, this time

starting from r/2 standard deviations from mean. Thus, we arrive at Table 4.3.

Since we will be evaluating (D(x) near x = 0, we should first Taylor expand this

126

Receive '0' (LED looks off) 0.5 - }b(-r/2) }I(-r/2)

Receive '1' (LED looks on) }A(-r/2) 0.5 - }((-r/2)

Table 4.3: Joint probability mass function (PMF) for communication with arbitrary

signal power L = rL,.

function around that point:

@(x) = @(x) L

= 0.5 +

-0.5+

d@(x)+ x + O(x 2)dx xo

d [ i1- - e x p

dx -_ 0 e1

exp (( 2

~2}dy] x + O(x 2)

x=O

-0.5+ +0(x 2 )

Substituting this expression for D gives us Table 4.4, which is valid only in the

low-power limit r < 1.

Send '0' (LED off) Send '1' (LED on)

Receive '0' (LED looks off) 0.25 + 1 r 0.25 - 1r

Receive '1' (LED looks on) 0.25 - 1 r 0.25+ Lr

Table 4.4: Joint probability mass function (PMF) for communication with low signal-

to-noise ratio r = L/L, < 1.

As with the 36 pW data point before, we can compute the contribution to the

mutual information I from the diagonal terms and off-diagonal terms separately, then

sum them. For convenience, let us define a new small parameter a = 1r.

127

Send '0' Send 'I' (LED on)(LED off)

X + 0(X22 X=O

The diagonal contribution is as follows:

Pxy(0, 0) - log (Px'Y(0, 0)QxY ( 0, 0)]

= (0.25 + a) log 0.25 a

= (0.25 + a) log (1 + 4a)

= (0.25 + a) - (4a - (4a)22

- (a - 22 +4a2 + O(a3 ))

In (2))

And the off-diagonal contribution is:

Pxy(0, 1) - log (Px'y (0, 1)\Qx'Y (0, 1)]

= (0.25 - a)

= (0.25 - a)

log 0.25-

log (1 - 4a)

= (0.25 - a) -(4a -4 ( a2

= (-a-2a2+ 4a22+ O(a3))

h (2) )e+( 2)In (2)J

Combining these terms, we get:

82Sa2 + O(a3 )in (2)a(4.6)

Thus we arrive at the general expression for maximum energy efficiency of com-

128

ln(2)± (3))

n(2)

a + In (2)

ln(2)+ O(a3))

n (2)

az2 + 0(ae3)

munication given a power measurement of the type from Chapter 3:

min - lim [" 2 (4.7)T r-+O

L2- Lo- At - 47rln(2) (4.8)

Lunity

In § 4.1.1 we saw that the 36 pW data point could be interpreted as communicating

about one bit per 1.7 pJ. If instead we had operated the LED at much lower power,

more efficient communication should have been possible.

Since from Table 4.1 we see that 36.336 pW of optical power could be generated

for just 1.7028 pW of electrical input power, we may use the q ~ 1/L scaling law

to infer that the power available at unity efficiency was Laity = 775.4 pW. (Note: a

best fit from the all of the above-unity efficiency data points [86] gives Luity ~ 764

pW, in good agreement with this figure.) Thus, using just the data from the second

row of Table 4.1, we find that the minimum work required per bit for this link is:

W _(3.214pW) 2

min - s - 47ln(2) = 103 fJ/bit (4.9)1 775.37 pW

For context, please note that the effective data rate for the channel at this operating

point is less than one bit per second. In § 4.3, we will address this issue using

orthogonal frequency-division multiplexing (OFDM).

4.1.3 Extrapolation to Carnot-efficient LEDs

As we discussed in § 2.4, in theory if all non-radiative recombination is eliminated

from an LED, but the device's active region remains optically thick, then the device

acts very nearly as a Carnot-efficient heat pump. Here we use the term "Carnot-

efficient" to mean thermodynamically reversible, without any entropy generation but

129

possibly with entropy transport, and as efficient as possible given the Second Law of

Thermodynamics. Recall now the results from § 2.3 and § 2.4. Since in such a device

the net amount of radiative recombination (i.e. recombination minus generation)

should still be given by Rrajiative = B(np - n?) = Bn?(eqV/kBT - 1), at low bias this

still contains a term linear in V. Thus we expect the idealized LED to have a finite

zero-bias resistance. Again, therefore we have:

L I 1 x -'- 1R q -Oc - (4.10)

IV 12 R ZB L

However, if we write 77 = Lusity/L, we must be careful to note that L..ity is defined

by the low-bias behavior; the actual unity efficiency point will be where V > kBT/q

and so will be much larger than Lusity.

With this in mind, we have performed a numerical calculation using Equation 2.24

for a diode with area A =0.0616mm 2 and red cutoff wavelength A = 2.6pm at temper-

ature T = 423K. We find that Lunity = 20.6 W/m 2 x A = 1.27pW, or about 1600x

larger than our measured result. Since the minimum work per bit scales inversely

with Lusity, our calculation suggests that for an LED with unity quantum efficiency,

roughly 63 aJ/bit should be required. Note that this is about 4 orders of magnitude

away from kBT ln(2) ~ 4 x 10-2 1 .

We know that in principle reductions in L, can also be made by decreasing the area

and temperature of the photo-diode. Consideration of such idealizations, however,

leads to an interesting question: What happens if we reduce the area and temperature

of the diode until L, falls by more than a factor of 125? Doing so would decrease the

work per bit to below kBT ln(2). At first this might seem possible because the the

input-referred current noise that leads to L, is inversely proportional to the photo-

diode's resistance, and the resistance across a p-i-n junction can be made extremely

large at cryogenic temperatures. An important hole in this analysis is that as the

130

thermal noise in the photo-diode is reduced, at some point other sources of noise in the

power measurement may dominate. In the following section, we consider a different

noise source: shot noise in the arrival of blackbody photons. In the experiments

we described in § 3.2, thermal noise due to lattice vibrations in the photo-diode

dominated over this contribution.

As we will see in § 4.2, the presence of this unavoidable noise source imposes a

lower bound on the work per bit required by a thermo-photonic heat pump. Although

we explore this problem theoretically, in principle it may also be measured experi-

mentally. We will return to this proposition as part of Future Work in Chapter 6.

4.2 Limits of Energy-Efficient Communication with

a Heat Pump

4.2.1 The Entropy Trade-Off

Consider a single photonic mode occupied thermally at a temperature To. Take the

expected number of photons in the mode to be E[N] = No. Now imagine that two

devices are capable of increasing the number of photons E[N] in the mode in two

different ways.

" Device A increases E[N] to No + 1 by deterministically adding a single photon

to the mode, making it a non-thermal distribution.

" Device B increases E[N] to No + 1 by increasing the temperature of the mode,

so that it remains a thermal distribution with some temperature T > To.

The photon number distributions for these scenarios are depicted in Figure 4-2. The

state of the photon field that results from the distortion by Device A is described

by a photon number probability distribution PN(n) which is the same as the original

131

distribution, except shifted to the right by 1. Since the entropy of a distribution (or

the von Neumann entropy of the corresponding density matrix) does not depend on

the labels of the outcomes, this state has exactly the same entropy as the original

distribution at To. Meanwhile the state of the photon field produced by Device B has

an increased expected number of photons by further spreading out the distribution.

This state has more entropy than the original distribution.

0.4 0.4 0.4

>0.3 E[3 0.3 1 [ I >0.3j [J3.go3E[N]= 3 .Z,. E[N]= 2 .go3E[N] = 3

S0.2 (Device A) 10 0.2 0 0.2 (Device B)

0 02

0.1 0.1 0.1

0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6Number of Photons N Number of Photons N Number of Photons N

Figure 4-2: Photon distributions for the initial thermal state (middle), as well as the

final non-thermal state produced by Device A and the final thermal state produced

by Device B. Both Device A and Device B increase the number of photons in the

mode, but only Device B increases the mode's physical entropy.

The extra entropy which Device B adds to the output state changes the lower

bound on how much work must be consumed imposed by the Second Law. The

Second Law does not permit the destruction of entropy, but since the photon field's

final configuration has more entropy, some of this entropy can in principle be drawn

from another thermal reservoir. If we presume the existence of another reservoir at

temperature To (in our case, the phonon bath), each bit of entropy AS which is drawn

from this reservoir brings with it energy TOAS. In essence, both devices are distorting

the initial state of the photon field, but the Second Law permits Device B to create

this distortion more efficiently. That is to say, in order to increase E[N] by 1, Device

A must consume hw of work, while Device B may consume less. Moreover, for very

small distortions of the original thermal state, Device B is effectively pumping heat

from some reservoir at To to a mode occupied at T = To + 6T. Here the Carnot

efficiency diverges: rCarnot T/6T -+ oo.

132

Now consider the possibility of using Devices A and B as transmitters, along with

a detector that counts the quanta of the photon mode in question, to form a simple

communication link. The fact that Device B can more efficiently increase the number

of quanta in the mode raises the interesting question of whether kBT ln(2) per bit

limit we derived in § 1.4 may be overcome by heat pumping.

In this section we will see that closer examination of such a link reveals a fun-

damental aspect of communication that we have thus far neglected. Device A and

Device B can both change the original occupation of the photon modes by adding the

same amount of energy, but the resulting final states are still different. In particular,

the final state that Device B produces is fundamentally less distinguishable from the

original state than the final state Device A creates. In essence, Device B makes effi-

cient use of the mode's capacity to store entropy with its efficiency improvements on

the transmitter side while Device A makes efficient use of that capacity to make the

final state more distinguishable on the detector side.

Thus the calculation of the efficiency limit for communication across a link made

with Device B is interesting not only as a generalization of a basic problem in commu-

nication and information theory, but suggests a fundamental connection between the

notions of entropy in thermodynamics and entropy in digital systems, a subject which

may become increasingly relevant as more efficient digital systems are developed.

4.2.2 Calculation of the kBTln(2) Limit

To calculate the theoretical minimum work-per-bit required to communicate with a

thermodynamic heat pump, we rely on two different bounds in combination:

1. The Carnot bound imposed by the Second Law of Thermodynamics. We use

the Second Law to place a lower bound on the amount of work required to pump

heat into the electromagnetic field at finite temperature.

133

2. The information-theoretic Shannon Limit derived from the Channel Coding

Theorem. We use the Theorem to place an upper bound on the amount of

information (measured in bits) which can be reliably sent across a noisy channel.

We begin by calculating the arrival rate of blackbody photons at a detector with

perfect quantum efficiency above it's band gap energy Egap and zero quantum effi-

ciency below it. For an incoming electromagnetic field occupied at finite temperature

T, we can calculate the number of above-gap photons per unit volume from the

density of modes in reciprocal k-space, the dispersion relation hw = hck giving the

photon energy in each mode with wave-vector k, and the Bose-Einstein distribution

giving the expected number of photons in each mode. Integrating over wave-vectors

corresponding to above-gap photon energies, we have:

N 2 d 3 k (4.11)

V (27)3 Egap/(hc) eXp ( 1

(kBTh) 3 J 2dx

72C3 Egap/(kBT) ex -

which leads to a particle flux of:

(kBTh) 3 f0 2dxJN 47 2 c2

fEgap/(kBT) ex - 1

From here we can simplify our calculation by assuming the above-gap photons are in

the dilute Boltzmann limit, so that:

JN=(kBT/h)3 oJN 4 2 2 (-I -e-x (X 2 + 2x + 2)) or (4.14)472 C2 Egap/(kBT)

JN (kBT/h) 3 E2 + 2x ± where = Egap4J c2 g2) kBT (4.15)

134

The number of above-gap photons incident on an illuminated detector of area A in

time At is therefore:

A = AAt(kBT/h) 3 (X + 2xg + 2) - ex9 (4.16)

This dimensionless number A can also be thought of as the number of photons oc-

cupying a finite volume of phase space. We note that these modes are in thermal

equilibrium by construction.

From this result, we can construct a cost function which represents the amount of

work required to pump heat from a reservoir at ambient temperature TA to a finite-

size system (i.e. the phase space volume flowing through the detector surface in time

At). As the energy and entropy from the reservoir are pushed into the system, its

temperature will rise. Since we are interested in the low-power limit, we consider the

case in which the temperature of the system begins at TA and ends at TB > TA.

The quantity of work required to pump each unit of heat into the system depends

on the temperature T of the system. For a Carnot-efficient heat pump, we may define

a function of temperature i(T) = dU/dW = T/(T - TA), and use it to express the

total work required to raise the system temperature from TA to TB:

fU(TB) dU fTdU T--T

W=]f dW = I = T - dT (4.17)U(TA) T(T) JTAdT T

135

The expression in Equation 4.16 can be differentiated to find the heat capacity dU/dT

of the small system. Keeping terms to leading order in xg > 1:

dU d- (AEgap)

AAt (kBT/h) 3=Egap - 4r2c2

AAt (kBT/h) 3

SEgap 47r2c2

~ AEgap = kB A X2

9 1 [(X2 + 2xg + 2) e~ ] + - AEgapTxg - j T

[xg (2xge-xg - X~e-xg) + 3 X2 eX9]

Thus since the heat capacity is finite, if we switch variables to T' = T - TA and only

keep terms to leading order in T', we may integrate to find:

W = JdW (L dU 1 T'

JTB-T dT TA±TdT'

- TB -TA [kB [xg(TA+ T' )] 2 A(TA ±T). TA T'] dT'

- [. T, T B T' d T

- k* [g(TkB g A) 0(T -i 2'TA+TBA T

SdTB -A)2=kB Ig A 2 TA 2

dU"~ {(TB - TA

-T=TA- TA.

In terms of the change in energy of the small system f dU = AU = AAEgap, we have:

W=AU TB -TA 1TA 2 (4.27)

or one half the typical Carnot expression for pumping heat between two reservoirs

at TA and TB. Intuitively this is because some portion of the heat that is pumped

into the small system is pumped across a small temperature difference compared with

TB - TA and some portion of the heat is pumped across almost the entire difference

136

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

(4.25)

(4.26)

TB - TA. Since we are considering small distortions to the equilibrium field (i.e.

we have linearized the energy in the small system as a function of temperature) the

average portion of heat sees half the temperature difference, or (TB - TA).

Using this result, we now look to construct an expression for W written solely

in terms of A. Since our measurements of the arrival of blackbody photons are over

time intervals ft much longer than h/6Ephton, where 6Ephotn is the range of photon

energies measured by the detector, the volume of phase space which we are probing

includes many longitudinal modes. The arrival statistics of the photons are therefore

given by the sum of many independent random variables representing the number of

photons measured in each mode. As a result, the arrival process of blackbody photons

counted by the detector is to good approximation a Poisson process; A is the expected

number of arrivals over time At. Because the probability mass function (PMF) of the

random variable N, which expresses the number of arrivals during a time interval At,

is more easily parameterized in terms of A than the photon temperature, we return

to Equation 4.16 to construct an expression for the work W in terms of A.

Differentiating A(T) and again keeping terms to leading order in xg:

dA 1 dU _ AEgap (4.28)dT EgapdT kBT 2

or equivalentlydA dT Egap (4.29)A T kBT

From this we can express the work W(A, AA) required to pump a volume of phase

space from a state with an expected number of photons A to a state with an expected

number of photons A + AA.

dA A EAA~- AT - (A4ga.3 TB -TA (4-0)

dT T=T, kBTA

137

Thus we have

kBTA AA 1 1 AA2

W(A, AA) = (AXEgap) AgaP - - kBTA - (4.31)TA 2 2 A

Now that we have a cost function, we must calculate the minimum cost W per bit.

This calculation was first performed numerically by the author (P Santhanam) and

analytically by Dr. Ligong Wang. Here we present both results.

Count m Photons (1 - a) e- a e(A+AA) (A+AA)mr

Count 2 Photons (1 - a) e-A a e-(A+AA) (A+AA)2

Count 1 Photon (1 - a) e-A A a e-(\+A) (A + AA)

Count 0 Photons (1 - a) e- a e-(A+AA)

Send Y=O (LED off) Send Y=1 (LED on)

Table 4.5: Joint probability mass function (PMF) for communication with Poissonsymbols.

Consider the following joint PMF fym(y, m), constructed using the probability

distributions in Figure 4-3 as conditionals, which represents communication over a

noisy channel consisting of a source which modulates the Poisson arrival rate of pho-

tons at a photon-counting receiver.

fYM(y, m) = oy, (1 - a) [+ eA] + -y,. a [ ((A +AA) e(A+AA) (4.32)IM!) M!

Here Y indicates whether a symbol with nonzero cost (i.e. the LED 'on' state) or zero

cost ('off' state) is sent by the source in a given time slice; M is the number of photons

counted by the receiver as a result. From the PMF in Table 4.5, it is clear that for a

small but finite AA, the bit of data encoded in Y cannot be reliably transmitted in

just one time slice. We may convey this fact by quantifying the maximum amount

138

5 10 15 20Number of Photons N

E[N] =X+AX= 12

5 10 15 20Number of Photons N

25 30

Figure 4-3: Poisson distributions with mean A (top) and A + AA (bottom). These

distributions form two conditionals, fm(mjY = 0) and fM(mIY = 1) respectively, of

the two-dimensional probability mass function fyi(y, m) representing the channel.

139

0.

4.

0

0.11-

E[N] = X = 10

0

0.21

25 30

CO

0CL

0. 11-

0

I W

of information (represented as a real, non-integer number of bits) which may be

communicated reliably per time slice over repeated experiments [50].

This fractional number of bits, the mutual information I shared across the channel,

may be calculated from the joint PMF in Equation 4.32. Using Equation 4.2, we can

write an expression for I in this problem:

I[fyM] = DKL(fNlfy - fM) (4.33)

where fy and fM are the Y- and M-marginal PMFs of fyu respectively. Again we

will take the logarithm in Equation 4.1 to be in base 2, so that I has units of bits.

From here we could in principle compute the marginal distributions from Equa-

tion 4.32 and find the maximum value of the ratio of I[fym] to the expected cost in

work &W(A, AA). This is the procedure followed in the numerical calculation. How-

ever, to expedite the analytical solution, we employ the general result from Verddi

[87] which states that when your codebook contains one zero-cost symbol (Y = 0)

and one nonzero-cost symbol (Y = 1), the channel capacity per unit cost (i.e. the

inverse of the minimum cost per bit) is:

D(fm(mJY = 1)HfM(mY = 0)) (4.34)c(Y = 1)

where c(Y = y) is the cost of the nonzero-cost symbol. In our case, the denominator

is our expression for the work W consumed during a time-slice in which the LED is

on'.

1 AX2Denominator = c(Y = 1) = - kBTA - (4.35)

2 A

140

3

2.5

u.20

C

o1

3:O.5

10 10-2 101 100 101AX (number of photons)

Figure 4-4: Results of numerical calculation of work per bit required for a Carnot-efficient heat pump. The solid red curve represents the work per bit as a functionof the average number of excess photons AA. (Note that AA is proportional to thepower of the optical signal.) The horizontal dashed line represents a consumption ofkBT In 2 of work, while the vertical dashed line represents a signal power equal to thenoise power (i.e. the signal-to-noise ratio is 1:1). Flash signaling (i.e. infrequent useof symbols with non-zero cost) was modeled using a = 10-4. Ambient temperaturewas taken to be 300K, the detector bandgap energy was taken to match Silicon at 1.12eV, and the detector area was taken as 1mm 2 . The time-slice duration of each symbolAt was taken to be 2.16 s so that the expected number of photons counted during anoff-state time-slice was very nearly 10, and the signal-to-noise ratio is ~ AA/10.

141

The numerator is:

Numerator0- ff (( ±))e( )lg[(A+AA)M) I~)

Y (A+ AA)M) e-(A+AA) .log M!M=0 M! (m -

2~ ((A+ AA)M e-(A+AA) M log 1 + A AAM=O

\0 ( + Am! e-(A+AA) -M log I + AA) AnA2.M=O

(A+AA)log1+ A In2

AA 2 A 1 -AA=(A+A A 2 A ) In2 1n2

1AA2 12- + -O(AA 3

2 A In 2

(4.36)

(4.37)

(4.38)

(4.39)

(4.40)

(4.41)

Combining these results, we arrive at an analytical solution to the question posed in

Equation 1.13. In the low-power limit where AA -+ 0, we find

(W'min bi = kBT In 2 (4.42)

This result matches the numerical results plotted in Figure 4-4.

4.3 A Thermo-Photonic Link

The calculations we presented in § 4.1 suggested that in the very low power limit,

in which the signal to noise ratio of the power measurements was less than 1, the

measurement represented the conveyance of some nonzero information (i.e. less than

1 bit) about whether or not the source was turned on. We then noted that if we

calculate the energy consumed to drive the source LED during the measurement, we

could find that the energy efficiency with which this "communication" was taking

place reached an asymptote at 103 femtojoules per bit. That is, in the same regime

142

where we found the Landauer limit in § 1.4, the minimum energy per bit found in that

calculation could also be found at low signal-to-noise ratio but had a higher value of

around 100 fJ/bit.

While 100 fJ/bit may be a reasonable energy efficiency for a low-power channel, the

corresponding effective data rate (i.e. the maximum rate at which it is theoretically

possible to use coding to communicate error-free [50]) of less than one bit per second

was highly impractical. However, because the aforementioned measurements only

used a very narrow band of frequencies of this linear channel, multiplexing could in

principle compensate for this. In fact, because the width of the frequency band which

the lock-in measurement uses is Af ~ 1/r, where r is again the integration time, a

densely frequency-division multiplexed channel can achieve the same symbol rate as

an OOK channel. Put simply, the rate at which O's and l's can be transmitted is

only constrained by the physical bandwidth which the hardware can achieve.

Here we describe a communication channel [88] constructed from an LED-photodiode

configuration closely resembling the setup from the power measurements. The hard-

ware control and data acquisition elements of the experimental channel were built

primarily by Duanni Huang, while the experimental design and execution were done

in collaboration with the author. The theoretical calculations used to extrapolate

from the final results were developed primarily by the author.

The first step to constructing a working channel was to replace the function gen-

erator current source and the lock-in amplifier with digital-to-analog (DAC) and

analog-to-digital (ADC) converters respectively. The ADC was chosen to have a high

sample-rate and bit depth to ensure the remaining hardware was functioning as an-

ticipated. A series of low-biased LED power measurements using a single modulation

frequency were performed with both the old and new hardware configurations; as seen

in Figure 4-5, measurements were in close agreement, indicating that the new hard-

ware had not introduced any large systematic errors and that the dominant source of

143

Copper LensHousing LED (1670C)

Photodioc

[ DAC-board0 Lock-in amplifier

--- - -- ---- -- - - - - - -S (k,T 1n2) Joules per photon

Ih.Unity wall-plug efficiency

X z 2.47sm %%qT = 1670C

,0 107

Output Light Power (W)

e (-20*C)

1

Figure 4-5: (top) A depiction of the hardware setup for the experimental link which

includes the components relevant for the transmission of the signal in the optical

domain. (bottom) A plot of LED wall-plug efficiency versus optical power. Measure-

ments taken with the lock-in amplifier and the analog-to-digital converter both match

the theory from Chapter 2 and each other quite well.

144

(a)

0

!F 10

0

-J

(b) -12

noise across the channel remained outside of the DAC and ADC elements.

During the operation of the communication channel, 2.5pim photons were emitted

by the LED at 167'C and detected by a photo-diode at -20'C with red cutoff wave-

length near 2 .6p-m. The observed quantum efficiency was 2xIO-I and the power at

unity wall-plug efficiency was 533 pW.

2:U

(a)'Pre-processing

16-bit OpticalI Encoder - 8-PSK H OFDM_ _D/A Channel

Phase 16-bit Gain* Bit Decoder FFT

Tracking ADStages

Post-processingL -I--------------- I-------------------------

10 10

(b ) . * ce** n(c)0i&_2 syymbolsj 2.5,

001044.110 1

E1 000 10*

-2 0.

-2 -1 0 1 2 3 995 1000 1005 1010 1015Real x 10" Frequency (Hz)

Figure 4-6: Subfigure a (top) is a block diagram of the experimental channel. Thediagram represents the flow of information from the input bit stream to the output bitstream. Subfigure b (bottom left) is a plot of complex amplitudes Bi which emerged

from the Fast Fourier Transform (FFT) block on the detector side of the channel. Theobserved points are clumped into 8 regions which correlate very strongly with the 3-bit sequence used to encode the relevant Fourier component. Subfigure c (bottom

right) shows the magnitude of the FFT of a different sample signal for which only 4of the 22 frequencies were intentionally excited. Because the time block over which

the signal was sent was 1 second, the frequency spacing in this plot is 1 Hz.

To create a low-power communication channel using the LED-photodiode pair at

low power, as seen in the block diagram at the top of Figure 4-6, we used a technique

145

5

1

5

known as Orthogonal Frequency Division Multiplexing or OFDM. In an OFDM chan-

nel, all of the information in a fixed-size block of data is sent simultaneously over a

fixed length of time T. During that time, one bit (or more generally a packet of bits)

is encoded into the amplitude of sine wave at with frequency f = 1/T which persists

over that block of time. The second bit is encoded into a sine wave with frequency

2/T and added to the contribution from the first sine wave. Since these two sine waves

are orthogonal over any interval of duration T in the sense that the normal notion of

the inner product of two functions is zero (i.e. when (f (t), g(t)) = J[ f(t')g(t')dt' = 0

we say f(t) and g(t) are orthogonal over [0, T]). In this way, the remaining bits in

the block of data are encoded into the higher harmonics, chosen to be orthogonal to

all the other sine waves in this interval. The resulting waveform, which contains the

information in the block of data, is used by to drive a current ILED(t) through the

source:M

ILED (t) = B3 l - Iocos (27fit + arg(Bi)) , (4.43)i=

where Bi is a complex number which encodes one or more bits into the ith harmonic

frequency, fi = i/T, where T is the analog block length, I0 is a coefficient with dimen-

sions of current which is varied in these experiments, M is the number of frequencies

multiplexed together, and we have used cos(.) instead of sin(-) here without loss of

generality owing to the variable phase of Bi. A plot of one such signal's Fourier trans-

form appears in the bottom right of Figure 4-6. In the example waveform, the time

block T is 1 second, so that the orthogonal waves are spaced 1 Hz apart. The Bi for

most of the frequencies in the plot is zero while the magnitude of the components at

4 of the frequencies take on the same nonzero value. Note that because the y-axis

is power, the plot does not show the phase information which is obtained in the de-

coding process. This phase degree of freedom is necessary for the codebook used in a

number of the experiments, which we explain next.

In order to further decrease the per-bit energy consumption of the link, in several

146

experiments we employed a form of phase-shift keying to encode three bits into each

of the {Bi} above. To do so, we designed our codebook to have 8 distinct symbols:

seven symbols were at the same amplitude but had equally spaced phases on the

interval [0, 2-r) while the one remaining symbol was taken as zero. The plot at the

bottom left of Figure 4-6 shows a series of measurements resulting from the use of all

8 symbols in this codebook with equal frequency- a choice we make for the practical

reason of wanting a simple mapping from bit sequence to symbol sequence.

It is worth noting that this codebook is counter-intuitive in light of the result

from [87], which suggests that the optimal codebook in the presence of a zero-cost

symbol involves only the zero-cost symbol and a single nonzero-cost symbol. A better

understanding of which assumption is invalid in our particular case remains a subject

of interest going forward. A working hypothesis is that both codebooks of the type

described here and those with a single nonzero-cost symbol converge to the same

minimum value for energy per bit in the low power limit (i.e. the solution is not

unique), but the optimal solution at finite bit rate in our channel reflects the specific

structure of our symbol space and cost function. Essentially, while any solution of

our form can be improved upon in the sense of having a lower cost, the practical need

for bit rate makes our solution preferable because it gives a constant factor increase

in rate with a cost increase that appears only at quadratic order in time-averaged

signal power.

The final technique we wish to discuss was developed to compensate for a frequency-

dependent phase lag which, because they only appear at higher frequencies, we suspect

originated from the detector-side amplifiers which were operating at high gain. To cor-

rect for this issue, we simply sent a pre-specified trial signal with nonzero-amplitude

symbols at every frequency (i.e. no frequencies were encoding all zeros), then cali-

brated out the effect by inverting this phase lag with a software phase advance just

after computing the FFT.

147

Once a bit sequence has been sent through the link, we characterize the fidelity of

the channel using the fraction of those bits which were incorrectly decoded- the Bit

Error Rate (BER). Once the signal is digitized on the decoder side, the signal is passed

through an FFT which generates a complex amplitude for each frequency representing

the real amplitude and phase of the relevant component of the signal's waveform. This

complex amplitude is then processed through a Maximum Likelihood (ML) decoder,

whose goal is to map that value to the most likely candidate symbol for its value

at the source. This task is greatly simplified because our protocol involves sending

each of the code-words with equal frequency. It is further simplified because the noise

source has a probability density which is a monotonically decreasing function of the

distance from the complex amplitude at the source (the noise distribution is Gaussian

empirically, which is to be expected because the noise is thermal and determined by

the temperature of the photo-diode lattice). Our ML decoder then has the following

simple geometric interpretation: we simply find the symbol whose complex value is

closest to the measured value in the complex plane.

Practical time constraints limited the number of bits we could test with any given

protocol, so in order to extrapolate to low BER values, we found it useful to also

model the errors. Furthermore this served as a model with which to compare exper-

iments. In our model, we assume the conditional probability distributions given a

symbol with complex amplitude Bk to be a two-dimensional Gaussian in the complex

plane, centered around Bk with some standard deviation ao. Since the noise is taken

to be additive, the conditional distributions for each symbol were taken as Gaus-

sians with the same standard deviation uo but centered on the point in the complex

plane corresponding to the source-side amplitude for that symbol. Combining these

conditional distributions with the uniform distribution of symbols emerging from the

source, we arrived at the joint PDF for our channel. We then used our simple geomet-

ric interpretation of the decoding process to segment the space of measured complex

148

amplitudes into nearest-neighbor regions surrounding each symbol and identifying the

probability for error given a particular source-side amplitude of Bk as the integral over

the regions which were outside the decoding region centered on the point Bk. The

overall expected error rate was simply the inner product of the vector of conditional

probabilities of error with the uniform symbol frequency distribution. Since this op-

eration only involved a single two-dimensional integral for each of the 8 symbols, it

was not computationally infeasible to model low values of BER; if we had elected to

randomly generate noisy measurements via a Monte Carlo method and decode them,

this would not be the case.

The final calculation relevant to the final results was the amount of energy con-

sumed by the channel. We operated the detector in photo-voltaic mode so that no

power was consumed in reverse-biasing the device. The power at the source was

calculated from a simple expression derived from Equation 4.43 as follows:

E -T2 1I2R M 1E = 1 \ ILED (t) 2 R dt' 2 -T.Z Bi| 2

, 444)

where R is the zero-bias resistance of the source LED. Note that the cross-terms of

the integral in Equation 4.44 disappear only when both of the following are true:

" the LED operates in the low-bias regime so that the current and voltage are

linearly scaled versions of the same waveform, and

" the waveforms at each frequency have no DC component, meaning that our

source is operating half in forward bias and half in reverse bias; the signal

remains visible by the linearity of the response in the low bias regime qV < kBT.

The results of our experiments appear in Figure 4-7. Our measurements indicate

that the channel was able to communicate using just 40 fJ/bit with a bit error rate of

3 x 10-3. As expected, decreasing the amplitude of the source waveform decreases the

energy consumption of the channel per bit transmitted Ebit, but also simultaneously

149

-.

04

0

0 , .

o 2.9kbps 8-PSKo 29kbps 8-PSK

88kbps 4-PSK---Theory 2.9kbps

-Theory 29kbpsTheory 88kbps

(a)10-12

10-1

10-2

10-3

10-4,

1

Current Device Exp.--Current Device Th.

1EQE = 1 andmatched detector area

a

LU

C:0

4-J0

- (b)10-20 10-18 10-16 10-14

Energy per bit (J)10-12 10-10

Figure 4-7: Subfigure (a) [top] plots the experimental results as paired values of Bit

Error Rate (BER) and energy per bit alongside theory curves using the signal-to-

noise ratio as a fitting parameter. The experiment labeled '4PSK' utilized a very

similar protocol to the '8PSK' experiments described in detail in the main text.

Subfigure (b) [bottom] shows the extrapolations of our model calculations based on

idealizations of the LED and photo-diode. The portion of the curves to the left of

the line denoting of kBT in 2 per bit do not necessarily represent values beyond the

Landauer Limit because at such high BER the amount of mutual information carried

across the channel by each '0' or '1' is significantly than 1 bit. The proximity of these

model curves to the kBT In 2 line, however, does suggest that further work on channels

using improved LED-photodiode pairs could serve as a platform for investigating the

thermodynamic limits of classical photonic communication.

150

100

CU

I-

+0iB

10-14 10 13

Energy per bit (J)

-16 10-15 10-11

100

101

102

103

CL)

0L_

LU4-J

C

I-

104110-22

- - - -

0

-

(b)

increases the probability that each bit will be decoded erroneously. Furthermore, the

relationship between Ebit and the BER is in good agreement with the model for the

most part. At high bit rates, where higher frequencies must be used, the presence

of greater noise at these high frequencies leads to increases in BER not captured by

the model here. We note that these deviations from the model are compatible with

originating in the limitations of the amplifiers used in our experiment rather than

inherent limitations of the LED-photodiode segment of the link.

The lower plot from Figure 4-7 shows extrapolations from these results under

idealizations similar to those we considered in § 4.1.3. The rightmost curve is the

result of the model calculation using signal-to-noise ratio as a fitting parameter. The

next curve to the left shows the results of a nearly identical channel, but using an

LED with 100% quantum efficiency. The leftmost curve represents the modeled results

given both 100% quantum efficiency from the source LED and a decreased noise level

from using a smaller-area photo-detector (matched to the emitter area) with the same

detectivity.

In spite of the very low quantum efficiency (2 x 10-) of the LED used as a source,

the channel described here saw performance around two orders of magnitude away

from that of state-of-the-art low-power laser communication channels using nano-

photonic techniques to minimize power consumption [89, 90]. While most of these

channels can transmit information with a higher maximum bit rate, our implementa-

tion is different because it does not require any fixed power consumption like a laser

which must reach threshold. As a result, we can transmit at low bit rates with low

energy consumption while other systems can only achieve low energy per bit when

both power and bit rate are far above the values reported here.

Although we have constructed a channel capable of genuinely transmitting data

with very little power at the source, some caveats apply. First, the source and detector

of the channel were not at the same temperature. However, the results of Chapter 2

151

and § 3.4 strongly suggest that this condition is not fundamental and that similar

results could be achieved using an isothermal configuration. Second, several aspects

of the channel's encoding and decoding were treated as exogenous for the energy

analysis; only the electrical power used to drive the source LED (recall that the

photo-diode did not consume power) was considered. In particular, the amount of

power required for trans-impedance amplification may be greater for a technique in

which the photo-current signal is very small. Nevertheless, for systems which need to

transmit data at kilobits per second with minimal power consumption on the source

side, this type of channel may be of practical interest. Furthermore, because this

channel takes a different approach to efficient communication which relies on efficient

photon generation, further experimentation may reveal new insights into the ultimate

limits of energy-efficient optical communication.


In this chapter we studied, both experimentally and theoretically, the minimum en-

ergy requirements for a communication channel whose source is a thermo-photonic

heat pump.

We began in § 4.1.1 by analyzing experimental results from Chapter 3 to determine

how much information about the state of the LED under test was being captured by

the detector circuit. We found that the amount of work consumed by the LED per

bit of information captured by the detector reached its maximum value in the limit of

low power. In this limit, we found that our existing setup could transmit information

for approximately 100 fJ per bit, and that such experiments on more ideal LEDs

could reduce this figure to less than 100 aJ per bit. In the case of an idealized low-

temperature detector, we found that the presence of thermal blackbody radiation

constitutes a source of noise with which any information-carrying optical signal must

152

compete. A theoretical analysis of this situation was presented in § 4.2 and showed

that the presence of thermal blackbody radiation emerging from the source (which

is required for LED heat-pumping) imposes a lower bound on the work per bit of

kBT ln(2).

We ended the chapter with § 4.3, in which we presented an experimental thermo-

photonic link capable of communicating at kilobit data rates while consuming just

40 fJ per bit in the source and detector diodes together. Extrapolations of this result

based on the physics in Chapter 2 and Chapter 3 suggest that it may be feasible to

develop a channel in which the dominant noise is from blackbody radiation. As LEDs

with high quantum efficiency at low-bias are developed, their use in such a channel

should enable communication with power consumption approaching the lower bound

imposed by the Second Law of Thermodynamics.

153


154

Chapter 5

High-Temperature mid-IR

Absorption Spectroscopy

In this chapter we explore the potential for using LEDs in the low-bias regime for high-

temperature infrared absorption spectroscopy. In § 5.1 we outline the motivation for

this work, focusing on the particular problem of developing a platform for the analysis

of the complex fluids found downhole in oil wells. In § 5.2 we connect the problem of

extracting spectroscopic information from a sample under analysis to the problem of

communication and make use of relevant results from Chapter 4. In § 5.3 we present

experimental data on high-temperature LEDs that constitutes a proof-of-principle for

using them for absorption spectroscopy, and subsequently evaluate the suitability of

these sources for a downhole spectroscopy system with specific targets. In § 5.4 we

explain the limitations of infrared photo-diodes caused by decreasing shunt resistance

with temperature. Finally, in § 5.5 we present results from an experiment in which

both the source and detector operate at high temperature and demonstrate that

thermo-electric pumping can be used to compensate for increased detector noise and

maintain signal-to-noise ratio at elevated temperatures.

155

5.1 Motivation

Thus far our study of thermo-electrically pumped LEDs has been motivated primarily

by scientific questions regarding the nature of the phenomenon and the constraints it

places on quantities of practical interest, such as power and efficiency. In Chapter 2

we saw that more power is available at a given efficiency when the ratio hw/kBT

is small. Furthermore, in Chapter 3 we found that hw/kBT ~ 10 characterized the

most experimentally-accessible regime for observing high efficiency LED operation. In

Chapter 4 we quantified the limits on information transmission that appear at the low

power levels where high efficiency LED operation is observed. Taken together, these

basic observations about thermo-electrically pumped LEDs point to applications of

LEDs at infrared wavelengths and high temperatures in which relatively low power

is required. One such application is downhole infrared absorption spectroscopy. Here

we pursue it primarily as an engineering problem.

Although the evaluation and extraction of crude oil from underground formations

benefits heavily from the in-situ analysis of the extracted material, the environment

downhole in an oil well is harsh and presents many simultaneous challenges [581.Downhole analysis systems must operate at temperatures ranging from colder surface

temperatures around 00 C up to 200'C and pressures up to 20,000 psi [91]. Moreover,

physical the size of the borehole limits not only the size of the platform, but also

the electrical power and communication bandwidth available to these systems. Nev-

ertheless, downhole fluid analysis systems have been recently developed by multiple

oilfield services companies and can provide valuable information from oil fields at

various stages of the extraction life cycle [58, 92, 93, 941.

Current downhole platforms like the one currently employed by Weatherford In-

ternational, Ltd. perform spectroscopy at visible and near-IR wavelengths, but do

not gather information from mid-IR wavelengths beyond about 2 Pm [59]. However

several valuable target analytes can be detected by their mid-infrared absorption,

156

including H2 S, C0 2 , and hydrocarbons of various lengths. Much of the reason for

the lack of mid-IR capabilities owes to the poor performance of both the sources (see

Figure 1-1) and detectors at these wavelengths and temperatures [59].

Our goal in this chapter is to assess the feasibility of using low-bias LEDs to add

infrared spectroscopy capabilities to an existing visible/near-IR downhole platform.

To do so we will first investigate the high-temperature operation of sources and de-

tectors, then later analyze them together to evaluate the feasibility of meeting the

constraints on temperature and power consumption faced by the existing downhole

spectroscopy platform.

5.2 Mapping Spectroscopy onto Communication

The purpose of a spectroscopy system is to extract information about certain prop-

erties of a sample or analyte. In the case of mid-infrared absorption spectroscopy,

the property of relevance is the absorption length of the sample under analysis for

photons in a certain band of mid-infrared wavelengths. Often the goal is to extract

this information as quickly and energy-efficiently as possible, much as the goal of a

communication system is for the receiver to extract the information contained in a

digital bit-stream from the transmitter.

Consider the logic of Figure 5-1. The top diagram is a simplified depiction of

an absorption spectroscopy system. We are free to think of the combination of the

photon source and sample as together as encoding the information of interest (i.e.

the sample's transmission coefficient) into the portion of the photon field carrying

light rightward out of the sample. Since the transmission coefficient is a real number

between 0 and 1, we are free to represent it in binary with finite precision. For

example, three-bit sequence '011' could refer to the interval of possible transmission

coefficients from 0.375 to 0.500, while the three-bit sequence of '100' could refer to

157

PhotonSample Detector

T = 37.5%

Transmitter Receiver

P = 0.375xP.

Digital Source which DigitalEncodes {0,1) * Power Receiver

r - P= xP--- -a P

0 11 I7xj P401

Figure 5-1: Depiction of the mapping between absorption spectroscopy and digital

communication. All three diagrams are meant to depict the same physical situation,

but described with different language. The top diagram is labeled as an absorp-

tion spectroscopy system. The diagram at the bottom left is labeled as an analog

communication channel. The diagram at the bottom right is labeled as a digital com-

munication channel. The information extracted by the user at the detector/receiver

side (i.e. the digital bitstream or transmission coefficient) is the same in each case as

well.

158

PhotonSource

the adjacent interval of 0.500 to 0.625. We note that in practical systems which

employ digital lock-in amplifiers to measure transmission of a sample, the bitstream

described here is closely analogous to the actual bitstream which results from the

discrete Fourier transform of the digitized photo-current signal.

From this type of mapping, we can make a few general observations. First, the

precision of our representation is determined by the length of the finite bit-stream.

If the transmission coefficient is known more precisely, more bits will be required to

represent it with that precision. Secondly, while in an information-theoretic sense,

each bit should carry the same amount of information, the leftmost bits are the most

significant bits and the significance of bits decreases from left to right. Thus if the

sequence of bits we use is longer than can be accurately decoded, then the rightmost

bits will not contain any real information about our quantity of interest.

The information-theoretic model of absorption spectroscopy outlined here may

help us debug real systems with multiple potential sources of noise. For example, if

an ensemble of measurements taken under identical conditions produces different bit

sequences for which the most significant bits are equally likely to contain errors as the

least significant bits, this may indicate the presence of electromagnetic interference

introducing noise in the electrical signal after digitization. If the ensemble produces

bit sequences which correspond to transmission coefficients that drift periodically in

time with the frequency of another environmental variable (e.g. the mechanical pump

frequency, the 60Hz wall power frequency), that may indicate an addressable design

flaw. If however the system is operating properly, we should see that the measured

bit sequences correspond to a transmission coefficient between far from 0 or 1, with

a standard deviation greater than the quantization limit.

Furthermore, this viewpoint combines with the intuition from Chapter 4 to offer

direction on future system designs. First of all, the model shows that the technique

employed in § 4.3 does not easily map to a technique for efficient low-power absorption

159

spectroscopy. If we use orthogonal frequency-division multiplexing to probe the sam-

ple over multiple lock-in channels simultaneously, we find that each frequency channel

detects a few bits of information, but that the channels contain duplicate informa-

tion. As a result, many such channels cannot be used to reconstruct the information

gathered by a single frequency channel with their combined power. If however, we

developed a technique by which multiple channels could acquire information about

bits with different significance, then their information could be used in this way. For

example, a setup using a bank of interferometers with different arm lengths could

be designed to probe changes in the real index of refraction of a sample at different

scales. An interferometer with a short arm would be sensitive to large changes in

index, while an interferometer with a long arm would be sensitive to small changes of

index but only able to measure the index modulo the amount required to change the

output intensity by one complete fringe. In this way, the information from different

levels of precision can be acquired simultaneously with parallel channels. Information

about the real index could then be used to find the imaginary index of refraction (i.e.

the absorption) via the Kramers-Kronig relations.

5.3 High-Temperature Sources for Spectroscopy

The same model of 2.1pm LED from § 3.2 was heated to 150'C, where above-unity

efficiency operation was easily detectable. Here, partially-absorbing Parafilm wax

paper samples were stacked between the emitter and detector, creating an optical

path with variable transmission. The results [86] are shown in Figure 5-2.

The measured transmission coefficients were in agreement with the Beer-Lambert

Law, indicated by the dashed fit line. These measurements offer confirmation that

sources operating in the / > 1 regime can produce sufficient signals to perform ab-

sorption spectroscopy at high temperature.

160

"*100

g 0

cn65-

CO)

~30 -

0 100 200 300 400Sample Thickness (jm)

Figure 5-2: Results of an absorption spectroscopy measurement performed on 1500 C

LED operating above unity efficiency. Note that transmission scales exponentially

with the thickness of the partially-absorbing material placed in the optical path as

expected. This represents confirmation that useful sample information may be ac-

quired in the above-unity efficiency emitter regime.

161

As discussed in § 5.1, three targets of interest for oil analysis in a high-temperature

downhole environment are the determination of absolute concentrations of CO 2 and

H2 S, and the relative concentrations of hydrocarbons at various lengths. All three of

these targets are potentially accessible via mid-infrared spectroscopy, if good enough

sources and detectors can be developed. CO 2 has a strong absorption feature near

A = 4.2pm, while H2 S has a strong absorption feature near A = 3.7pm. LED sources

at both of these wavelengths have been developed in the InAsSbP:InAs material

system. As seen in Figure 5-3, sources at 3.7pum have been tested at high temperature

and exhibit the same temperature-dependence as other thermo-electrically pumped

LEDs: their low-power wall-plug efficiency increases with temperature. The efficiency

at fixed output power is more than 10x greater at 100'C than at 25'C. As evident

from Figure 5-3, this increase is more than sufficient to compensate for the spectral

redshift up to this temperature and beyond.

Due to the common presence of Carbon-Carbon single and double bonds and

Carbon-Hydrogen bonds, many hydrocarbons absorb infrared light around A = 3.4pm.

However, the shape of the absorption line in this range differs between chains of dif-

ferent lengths because of the different distributions of these types of bonds [95]. As

a result, sufficiently sensitive spectroscopic analysis of this absorption line can be

used to "fingerprint" a mixture of hydrocarbons to find their relative concentrations.

The capability to perform such analysis in the harsh downhole environment could

prove industrially applicable by improving estimates of oil-to-gas ratio for valuation

of existing wells and feedback into geophysical models used to plan extraction oper-

ations. The LEDs examined in § 3.4 emit at the relevant wavelengths and could be

suitable for a high-temperature downhole system since they exhibit this same type of

improvement with temperature as well.

The results of initial experiments on existing crude oil samples are shown in Fig-

ure 5-4. At left, Fourier-Transform Infra-Red (FTIR) spectroscopy reveals a clear,

162

1000c0

250C

0 "

0-6 10-5 10-4 0-3 0-2 0-110 10 10 10 10 10

Voltage (V)

3.6811m3.41pm .91Pm

..~~ ~ .. ...

3 3.2 3.4 . 3.8 4 4.2Wavelength (pm)

4.2

E 4

3.8

C3.6

3.

1 40 60 80Temperature (C)

Figure 5-3: Data for an InAsSbP:InAs light-emitting diode. Top: Output Power ver-

sus Voltage at two temperatures. Bottom left: Emission spectrum at 25*C. Bottom

right: Estimate of spectrum's redshift with temperature. The LED delivers signif-

icant optical power around the 3.7 pm H 2S line across the range of temperatures

investigated.

163

-6

10-10-

0-~ -9

10

100

10~11

0.6

0.

0.4

C

Ceted

Bnm/K

~3nm/K100

spectrally-isolated absorption line from a 1Optm-long optical path through a sample

of crude oil. This short attenuation length renders conventional optical geometries

impractical due to the presence of particulates in crude oil. Instead, an Attenuated

Total-internal-Reflection (ATR) geometry is being considered, and a prototype in-

strument is currently in development. At right, significant differences between two

samples from different oil wells can be resolved. The clarity of these differences

suggests that fingerprinting with sufficient accuracy to resolve typical hydrocarbon

length distributions in crude oil should be possible with just a handful of filtered

optical channels. We note that these measurements were made with the oil samples

at standard temperature and pressure, but blurring of these lines at high tempera-

ture and pressure [95] could present a significant challenge for any type of downhole

hydrocarbon fingerprinting.

006

WWavelength (am)

Figure 5-4: Left: Fourier-Transform Infra-Red (FTIR) spectroscopy reveals a clear,

spectrally-isolated absorption line from a 10 pim-long optical path through a sample

of crude oil. Right: FTIR spectroscopy of two crude samples from two different wells

reveals significantly distinguishable line shapes in this range.

5.4 High-Temperature Infrared Photo-Detection

As noted by Fujisawa, et. al. in Ref. [59], poor room-temperature performance

of mid-infrared photo-detectors compared to near-infrared and visible, and further

164

degradations in signal-to-noise ratio of these systems with increasing temperature

often prohibit downhole analysis systems from using mid-infrared spectroscopy. This

decreased performance is captured by the decrease in shunt resistance (i.e. zero-bias

resistance of the photo-detector) with temperature, as seen in the I-V characteristics

of the photo-diode shown at left in Figure 5-5.

This decrease is fundamentally due to the increase in concentration of thermally-

excited carriers in the diode. This increase in carrier population leads to increased

recombination at a given voltage, and ultimately more current flow. Note that this

physical explanation is in close analogy with the decrease in zero-bias resistance of

an LED; this forms the basis of the emitter-detector compensation concept described

in 5 5.5.

120

1001.5 --

1 .. LED (Emitter)

0.5 .

250K.O ~40-

-0.5 -. .-

-1 350K .. 20 Photo-Diode a-1.5 .(Detector)

-0.1 -0.05Voltage (V) 0 0.05 500 310 320 330 340 350Diode Temperature (K)

Figure 5-5: Left: Current-voltage characteristics for a HgCdTe p-i-n photo-diodeat various temperatures. The detector's cutoff wavelength ~ 6pm. Note that theslope of the I-V curve around the origin (i.e. R- ) increases with temperature.Right: Shunt resistance of a photo-diode and zero-bias resistance of an LED withtemperature. Both decrease exponentially with temperature. The LED's emissionwavelength is ~ 3.7pum.

We have constructed a model for the noise in our photo-detection circuit. It is

depicted as a circuit diagram in Figure 5-6. In this model, we include two sources of

current noise, one from thermal vibrations in the diode and one from shot noise, in

parallel with the resistance of the diode at the input to the trans-impedance amplifi-

165

Photo-diode

I II I

shot0 I

:R B I

/

I - I

I + II

Trans-ImpedanceAmplifier (TIA)

Figure 5-6: Circuit diagram representing the basic noise model developed for our

photo-detector circuit. Here RZB stands for the zero-bias resistance of the photo-

diode (or equivalently the shunt resistance), Vth stands for the zero-mean voltage

noise generated by thermal motion of electrons within the photo-diode and the metal

contacts and wires which connect the diode to other circuit elements, I, stands for

the zero-mean current noise due to shot noise in the incident photon flux, and RGain

stands for the trans-impedance gain of the combined amplification stages measured in

Ohms (Q). Note that we have omitted the analog-to-digital conversion in our model

because we assume any noise introduced at this stage is negligible compared to the

other noise sources included in the model.

166

cation (TIA) stage. Since we are interested in the outcome of a lock-in measurement,

we would like to express the root mean square -)rms of the Johnson-like thermal volt-

age noise [96] (Note: we use the term "Johnson-like" because Johnson noise typically

refers to noise from a resistor rather than a diode) in terms of the bandwidth Af as

follows [97].

KVthrms = 4 kBT AfRshunt

(5.1)

(Vth)rms = 4 kBT Af shunt

For lock-in measurements with a pass-band width of 1 Hz (the time constant

T = 1 second) and responsivity RPD = 2.96 A/W, we then expect the standard

deviation of our lock-in measurements at low detector temperatures to correspond to

a noise equivalent power (NEP) for this noise source of:

V4kBT AfRshunt --NEPth = x PD)

(5.2)

x (2.96)- = 5.64 pW59.49

Likewise if we perform the same calculation using the absolute temperature, zero-

bias resistance, and responsivity of the photo-diode across the range of temperatures

tested, we can find the thermal contribution to NEP. In Figure 5-7, the results of

these calculations are presented beside experimental data.

Also included in Figure 5-7 is an estimate of the noise caused by thermal photons

from the 300K ambient environment being absorbed by the photo-diode. When the

temperature of the photo-diode is above 300K, the interactions between the incident

thermal photon field and the photo-diode's active region doesn't serve to create a

significant non-equilibrium population of free carriers, so we should not expect this

167

102Standard Deviation

in Power Measurements

L

0 10

0.

Johnson-like Shot Noise fromJoTh nsr lie Ambient Photons

10 Thermal Noise

210 240 270 300 330 360Temperature (K)

Figure 5-7: Noise-equivalent power (in pico-watts) of the lock-in measurements of

optical power emitted from a mid-infrared LED and detected by a photo-diode ver-

sus the temperature of the diode. Note that the thermal and electrical conditions

of the source were constant across these measurements. The hollow square mark-

ers represent experimental measurements; the hollow circles represent a model of

the noise-equivalent power from Johnson-like thermal noise which uses experimental

measurements of the photo-diode's shunt resistance; the dark dotted line represents

a model of shot noise in the incident photons in the limit of low detector tempera-

ture. The shot noise calculation assumes the photo-current from incident photons is

dominated by those produced by the 300 K ambient in the laboratory. Qualitative

agreement is reasonable, but observed levels exceed the modeled values by about a

factor of 3. Likely sources of model errors include constant factors from the digital

signal processing of the lock-in amplifier, uncertainty in the responsivity of the photo-

diode near its cutoff wavelength of 6 pm, and the omission of other possible noise

sources caused by the trans-impedance amplifier.

168

noise source to be relevant for detector temperatures above ambient temperature.

Below ambient temperature, however, incident thermal photons within the responsive

band of the detector can generate electron-hole pairs in excess of the equilibrium

population that are swept out and appear as photo-current.

Because the radiation in the responsive band is strongly multi-mode on the timescales

of our measurements, we begin our analysis with the assumption that the arrival of

thermal photons behaves as a memoryless Poisson arrival process. Based on the de-

tector's data sheet [98], the detector used here had an effective area of 1mm2 and

was responsive out to 6pm. A quick calculation of the equilibrium blackbody flux

through this area of photons with wavelength 6pm and shorter yields about 16pW. If

we assume this flux is dominated by photons near the 6pm edge, and we use the de-

tector's responsivity to these photons is perfect (RPD = 4.84A/W), this corresponds

to an arrival rate of about A =4.9x10" detected photons per second.

We now use the basic result that the time-derivative S of the random variable N

representing the accumulated arrivals from a Poisson process has an autocorrelation

whose Fourier transform is flat with a value of A as a function of angular frequency

w away from base-band. We find that the "power spectral density" of the noise

integrated over a band Af is 2 x 2wRA x Af, where the extra factor of 2 accounts for

only consider positive frequencies f. Because A has no units, we should be careful

with the units in this expression. Note that the "power spectral density" in this case

refers to the absolute square of the Fourier transform S(f) of the time-dependent

random variable S(t), which itself has units of photons per second. To find the units

of S(f), we find that the units of the "signal energy" f IS(t')J2dt' are photons 2 per

second. Parseval's relation then indicates that the "signal energy" calculated in the

Fourier domain, Af IS(f') 2df' = 47rA - Af, has units of photons2 per second as well.

As a result S(f) has units of photons. Including just the Fourier components within

the 1 Hz band of our lock-in photo-current measurement, we should therefore expect

169

the "signal energy" of a time interval of 1 second to be 47A or 6.2 x 1015. We therefore

conclude that the effect of ambient photons on the cooled detector should be to add

current noise Ishot with zero mean and (fshot)rms = 12.6 pA.

Since this root mean square current fluctuation is interpreted as noise in the arrival

of 3.7pm photons, to convert this figure back to noise equivalent power to compare

with experimental measurements, we must now use the same responsivity we assumed

when converting our photo-current measurements back to power originally (RPD =

2.96 A/W). We therefore arrive at an estimated noise equivalent power for the shot

noise due to ambient photons of:

NEPshot = 4.26 pW . (5.3)

As seen in Figure 5-7, this figure is roughly 3 times smaller than the average

uncertainty in our low-temperature power measurements at -13'C and -53 0 C of about

12 pW. The preceding calculation was essentially an estimate, and because of its

sensitivity to certain parameters like the cut-off wavelength of the detector and the

ambient temperature as seen through the acceptance cone of the detector's immersion

optics, we should not expect very high accuracy. Based on the detector's data sheet

[98] we estimate the width of the Urbach tail to be around 30 meV; an uncertainty

in the cutoff wavelength of this magnitude would change our estimated NEP by

approximately a factor of 1.5. Although the ambient environment in the laboratory

was roughly 300K, it is also plausible that thermal radiation emitted by the heat

sink on the heat rejection side of the photo-diode's thermo-electric cooler could be

dominant; an increase in the temperature of incident radiation by 5 K would increase

our estimated NEP by 15%.

Furthermore, a more careful calculation may consider the following effects: con-

stant factors in the noise bandwidth for a given time constant from the lock-in ampli-

170

fier's digital signal processing (we assumed Af = 1/T where T is the time constant),

the effects of wavelength-dependent responsivity with special attention to the band-

edge of the detector, and the finite acceptance angle of the immersion lens abutting

the responsive area of the photo-diode. Considering the uncertainties in the input

quantities, the author believes the incidence of thermal photons within the responsive

band of the detector cannot be excluded as the dominant noise source in the mea-

surements with detector temperature < 300K. Further experimentation and modeling

could yield a more complete analysis of the noise in our power measurements, and

may in fact be a necessary step for building a spectroscopy system at this wavelength

which is designed to operate in a high temperature environment. We will return to

this topic, as well as the potential relevance of ambient thermal photons in the context

of communication with a heat pump, in our discussion of future work in Chapter 6.

5.5 High-Temperature Emitter-Detector Compen-

sation

Although the increase in noise associated with decreased photo-diode shunt resistance

is a robust consequence of operating in an elevated temperature environment, the

logic of Chapter 2 offers an equally robust mechanism to compensate for this. As

the resistance of an LED around the origin RZB,LED decreases with temperature, if

the emitter's quantum efficiency fEQE remains fixed a given voltage results in more

proportionally more light emission in the low-bias regime. Since the signal at the

photo-detector is proportional to the light output from the LED, we see that the

signal Vsignal emerging after trans-impedance amplification also scales inversely with

RZB,LED:

Vsignal = RGain ' RPD ' 7EQE -RB,LED . VLED - (5.4)

171

Although the photo-diode's responsivity RPD and the LED's quantum efficiency

7EQE both change significantly with temperature, we will primarily focus on the

temperature-dependence of the RLED term here. In our present experiment as

the detector temperature is raised from 300 K to 350 K, the responsivity falls by a

factor of 3 and the quantum efficiency falls by just over 10%. Meanwhile the LED's

zero-bias resistance decreases by a factor of 9.2.

By comparison, if the noise in our measurements above ambient temperature is

dominated by Johnson-like thermal voltage noise, then the standard deviation in our

lock-in measurements of the voltage signal after trans-impedance amplification VNoise

can be expressed as follows:

VNoise - RGain ' V4 kBT Af (5.5)s/shunt

As the temperature is raised from 300 K to 350 K, the explicit temperature depen-

dence in this expression increases by 17% while the square root of the inverse shunt

resistance increases by 42%.

Combining the expressions for Vsigna, and VNoise from Equation 5.4 and Equa-

tion 5.5 respectively, the signal-to-noise ratio (defined here as the ratio of the stan-

dard deviation to the mean of a lock-in optical power measurement with 1 Hz of noise

bandwidth) can be expressed in a way that reveals its temperature dependence.

SNR __ Vsignai _ RPD 7 EQE VLED /Rshunt (5.6)VNoise /4 kBT Af RZB,LED

We have arranged the above equation so that the dominant temperature-dependent

terms (i.e. the resistances of the source and detector diodes near zero voltage) appear

at the end. Recall now that the resistance of a diode decreases exponentially with the

ratio of the thermal energy kBT to the bandgap energy Egap (i.e. oc e-Egp/kBT). If we

assume the bandgap of both the source and detector diodes are similar in magnitude,

172

we find that under conditions of fixed source side voltage amplitude the SNR of

the combined system actually increases with temperature. Note also that at shorter

wavelengths, the diode resistances become exponentially more sensitive to changes

in temperature, while the competing effects of reduced responsivity and quantum

efficiency may not. Thus our model suggests that this low-bias spectroscopy system's

signal-to-noise ratio should increase with temperature because of the combined effect

of both diode resistances, and that this should remain true for similar near-infrared

systems as well.

As the resistance becomes small enough, the voltage noise at the input of the trans-

impedance amplifier could replace the thermal Johnson-like noise. Let us now consider

the temperature-dependence of the SNR when this noise source is dominant. The

relevant circuit diagram for this noise model is identical to the one in Figure 5-6 except

with the thermal voltage source replaced by one with a magnitude (VTIA)rrns which

is determined by the first stage of the trans-impedance amplifier. The corresponding

expression for the noise in the final measurement is:

VNoise RGain VTIA)rms (5.7)Rshunt

and the signal-to-noise ratio is:

SNR - Vsignai = RPD - EQE * VLED X shunt(5.8)VNoise RZB,LED

In this case, the scaling of SNR with the diode resistances depends equally on both

diodes. However, since Egap for the emitter must be at least as large as Egap for the

detector in order for the emitted photons from the LED to fall within the responsive

band of the photo-diode, the oc e-Eap/kBT scaling of the resistance is more sensitive

for the emitter. Thus even when the TIA's voltage noise is dominant, the SNR of the

combined system should increase with temperature.

173

Finally, we consider the case where current noise is dominant. Whether the dom-

inant current noise is from the shot noise in the incident thermal photons or results

from the first stage of the TIA, the shunt resistance in our model does not affect the

noise in the final measurement:

VNoise = RGain - (In)rms , (5.9)

and the increased signal strength again dominates:

SNR = Vignal - RPD 77EQE VLED 1(510)VNoise (In)rms RZB,LED

To summarize, we have presented a model for the signal-to-noise ratio of a mid-

infrared absorption spectroscopy system which uses a small AC voltage to drive an

LED and performs a lock-in measurement on the amplified photo-current signal from

an unbiased photo-diode. By assuming that the temperature dependences of the

source LED's quantum efficiency 7EQE and the detector's responsivity RPD are neg-

ligible compared to the temperature dependences of the zero-bias resistance in either

diode, we found that the combined system's signal-to-noise ratio should actually im-

prove with temperature. This result holds whether the dominant noise source in the

system is the Johnson-like thermal noise in the photo-diode, the shot noise of incident

thermal photons, or either current- or voltage-noise from the input stage of the TIA

which provides trans-impedance gain to the photo-current signal.

We have also performed an experiment to test this hypothesis. A basic absorption

spectroscopy system was built using an LED emitting at 3.7pm (see Figure 5-3 and

related discussion) and a photo-diode responsive to light at wavelengths from 2 to

6 pm (see § 5.4) and tests were performed at various temperatures for both devices.

To isolate our measurements from effects related to the red-shift of the source and the

wavelength-dependent transmission of a sample, no sample was introduced into the

174

TLED Tphoto-diode

0 2- 10 Rshunt' PD'

& ZB,LED

C/)

Z

0 1 TLED fixed at 300K10, L10

CoRshu and RPD

10220 240 260 280 300 320 340

Photo-Diode Temperature (K)

Figure 5-8: Signal-to-noise ratio of a basic low-bias lock-in mid-infrared spectroscopysystem versus photo-diode temperature. The hollow red squares indicate measure-ments in which the LED source was held at 300 K; the hollow blue circles representmeasurements in which the LED source was matched to the photo-diode tempera-ture; the black dotted line represents a model in which the source is fixed but theshunt resistance and responsivity of the detector take experimental values; the greenlong-dashed line represents a similar model in which the zero-bias resistance of thesource LED also takes on values from experiments with the source LED at elevatedtemperature. Note that no sample was placed between the source and detector forthese measurements.

175

optical path for these tests. The results appear in Figure 5-8 and confirm empirically

that the SNR of the combined system does in fact increase as the temperature of both

the source and detector diodes are simultaneously increased from 300 K to 350 K. In

a practical high-temperature spectroscopy system, increases in signal-to-noise ratio of

this type may allow compensation for other issues, such as red-shifting of the source

away from the target wavelength, which are outlined in § 5.3. We briefly discuss the

practical potential of this type of spectroscopy system in Chapter 6.


In this chapter we have conducted experiments on a basic absorption spectroscopy

system implemented using a 3.7pm LED driven by a small AC voltage, a photo-diode

sensitive from 2 to 6pm, a trans-impedance amplifier, and a digital lock-in ampli-

fier. In keeping with the observations of Chapter 3, the decreased performance of the

source-side LED at conventional operating points (where qV is on the order of Egap)

is reversed in the low-bias regime. In the low-bias regime, the increased output power

of the LED at constant input voltage is shown to be sufficient to compensate for the

decreased performance of the detector photo-diode at elevated temperatures. Models

are developed for the temperature-dependence of the noise in the detector circuit and

reasonable agreement with experiments is observed. The noise models on the detec-

tor side are then combined with the LED models from Chapter 2 to create a larger

model for the spectroscopy system's overall signal-to-noise ratio. These models sug-

gest that for a variety of potential noise sources, the improvements on the source-side

should outweigh the decreased performance on the detector-side, leading to a signal-

to-noise ratio which increases with temperature. From this work we conclude that

the exponentially decreasing performance with temperature of mid-infrared LEDs

and photo-diodes at conventional operating points may not, as previous authors have

176

suggested [59], prohibit the development of mid-infrared absorption spectroscopy sys-

tems capable of operating in high-temperature environments provided they employ a

zero-bias lock-in photo-detection technique along the lines described in § 3.1.1.

177


178

Chapter 6

Conclusions and Future Work

In this chapter we present a high-level summary of the work detailed in this thesis,

then using these ideas we describe several potential research directions going forward.

In § 6.1 we combine the results of Chapters 2 and 3 to construct a physical picture

of optoelectronic device operation, and apply that thinking to understand the results

from Chapters 4 and 5 regarding communication and spectroscopy respectively. Our

subsequent discussion of related research directions, some of which the author expects

to actually pursue in the near term, will be organized as follows. In § 6.2 we will

outline some questions of scientific interest which have been raised in the course of

this work. In doing this, we will first focus on problems associated with physical

entropy and information in photonic systems followed by problems related to the

physical entropy and information of electrons in semiconductors. In § 6.3 we describe

a number of future directions for applied work, some of which we have alluded to

in Chapters 2 through 5. Finally, in § 6.4 we discuss the long-term prospects for

energy-efficient solid-state lighting and photonic communication by considering the

constraints imposed by the Second Law of Thermodynamics in light of this work.

179

6.1 Thesis Summary and Conclusions

In this thesis we have presented theoretical and experimental results in support of a

thermodynamic interpretation of incoherent light generation by light-emitting diodes.

From the thermodynamic analysis of charge and entropy transport in a forward-

biased diode from § 2.1, we found that the carrier injection process necessarily requires

the absorption of lattice heat through the Peltier effect whenever the bias voltage qV

is less than the bandgap energy Egap. In § 2.2, we directly computed the entropy

removed from the electron-hole system by a single radiative recombination event. By

dividing the energy of the resulting photon by this quantity, we arrived at a simple

expression for the effective temperature T* seen by inter-band processes in terms of

the voltage V, the photon energy hw, and the lattice temperature Tiattice:

T*Tlattice I - . (6.1)

Using an analogy common in statistical mechanics, we saw that T* serves as a sort

of "exchange rate" between entropy and energy for distortions of the electron-hole

system which conserve charge. From the Second Law constraint disallowing the dele-

tion of entropy, we saw that T* serves as an upper bound for the temperature of the

outgoing optical field, and thus an upper bound on optical spectral power density. In

practical terms, Equation 6.1 connects voltage directly to brightness.

In § 2.3 we considered non-ideal LEDs, including those with low external quan-

tum efficiency. We found that although these devices could not achieve net electro-

luminescent cooling at high bias, at very low voltages they could. In fact, the linearity

of the diode's response to application of a small bias voltage showed that in theory

every light-emitting diode should experience cooling at sufficiently low voltage. How-

ever, since this low voltage constrains the outgoing optical field to a temperature

barely above ambient, observation of this phenomenon requires measurement of very

180

small amounts of light.

Next, in § 2.4 we saw that since a device with no non-radiative recombination

has no sources of irreversible entropy generation, as an LED's quantum efficiency ap-

proaches unity it behaves increasingly like a Carnot-efficient heat pump. Nevertheless,

since there is a fundamental limit on spectral power density imposed by the Second

Law limit on the maximum temperature of outgoing photons, there is a Carnot bound

on efficiency at fixed power density for an LED with a known low-voltage emission

spectrum.

The preceding results substantiate our interpretation of an LED as a thermody-

namic heat pump. They moreover justify two basic counter-intuitive aspects of the

thermal physics of highly efficient LEDs at voltages V < Egap/q:

" Instead of discharging waste heat into the device's lattice, an efficient LED cools

its lattice by pumping heat into outgoing photon modes which carry the energy

(and entropy) away from the device.

" Since heat can be pumped with a higher coefficient of performance against a

smaller temperature difference, an efficient LED source with a given absolute

spectral intensity (which determines the outgoing photon temperature) becomes

more efficient in a higher-temperature environment.

As a first step in developing devices closer to the Carnot limit, in § 2.5 we used

an experimentally validated computational model of a 2.15 ym InGaAsSb emitter

to design a new layer stack for operation at sub-bandgap voltages. We found that

significant improvements should be attainable using existing technology and that the

optimized device's behavior should exhibit a monotonic power-efficiency trade off

qualitatively resembling that of a Carnot-efficient device. In § 2.6, we closed our

theoretical discussion of thermodynamic device behavior by noting that analysis of

this nature is quite general and that in fact the flow of electrical current through

181

any semiconductor device (or combination thereof forming a closed circuit) can be

analyzed as a closed thermodynamic cycle.

In Chapter 3, we presented experimental evidence to test these theoretical predic-

tions. We began by providing a short description of some major techniques required

for the efficiency measurements that followed. These included the use of an AC LED

drive current with phase-locked photo-detection, feedback thermal control with lim-

ited temperature slew rates to avoid thermal shock, and some basic optical design

for efficient collection of photons from an imperfectly collimated LED source. In

§ 3.2 we presented the first demonstration of electroluminescent cooling by observing

electrically-driven optical power in excess of the electrical power required to drive a

2.5 pm LED at 135'C. Next, in § 3.3 we documented an unsuccessful attempt to make

a similar observation from a 4.7 Mm LED. The experiment was expected to achieve

unity efficiency at higher power density due to the longer emission wavelength, but

increased non-radiative recombination, leakage, and contact resistance offset the an-

ticipated increases. In § 3.4.1 we used LEDs emitting at 3.4 Pm to obtain further

evidence that the optical power measurements found throughout this chapter were not

the result of linear emissivity modulation. Finally in § 3.4.2 we present observations

of LEDs at 3.4 and 4.7 pm operating above unity wall-plug efficiency.

In Chapter 4 and Chapter 5 we explored the application of efficient LEDs at

low forward bias to low-power digital communication and high-temperature infrared

absorption spectroscopy respectively.

In § 4.1 we motivated our consideration of these LEDs as a source for optical

communication by reinterpreting the power measurements from Chapter 3 as com-

munication over a very slow channel. For existing measurements of a 150'C LED

emitting 2.5 Mm photons onto a 3mm-diameter photo-diode at 25'C, the correspond-

ing channel required just 1.7 pJ per bit. Our subsequent discussion of extrapolating

these measurements to the low power and high quantum efficiency limits indicated

182

that future experiments could approach the well-known Landauer limit [48, 49, 52]

of kBT In 2 per bit. In § 4.2 we analyzed an idealized system with no sources of irre-

versibility and no noise other than the presence of equilibrium blackbody radiation

at the optical frequencies which contain the signal. Using a codebook which was

optimized for energy efficiency, we solved for both the mutual information shared

across the channel and the work required to generate the signal in the low power

limit. Using both analytical and numerical methods, we found that in this limit, the

work required by this idealized channel is exactly kBT In 2 per bit. Finally in § 4.3 we

investigated orthogonal frequency-division multiplexing as a means of increasing the

data rate of such a channel without sacrificing its per-bit energy efficiency. We de-

scribed our experimental realization of a multiplexed 3 kbps low-biased LED channel

which consumed just 40 femtojoules of electrical energy per bit with a bit error rate

of 3x10 3 .

Our discussion of the potential for thermo-electrically pumped LED sources in

spectroscopy began in § 5.1 with a brief discussion of the technological need they

could fill. We saw that while applications such as downhole fluid analysis in oil wells

and combustion exhaust gas analysis possess valuable spectroscopic information at

mid-infrared wavelengths, the inefficiency of existing sources and detectors at room

temperature and above are often prohibitive. However, since the low-biased LEDs

discussed in this work are highly efficient and can become more efficient at higher tem-

perature, they could be used for absorption spectroscopy in these high-temperature

applications. After a brief aside in § 5.2 to establish a framework to analyze a spec-

troscopy system using the same information-theoretic tools as in the previous chapter,

we looked more closely at the high-temperature behavior of LEDs and photo-diodes

in § 5.3 and § 5.4 respectively. We found that while the performance of the photo-

diode-based detector circuit decays exponentially with temperature, the performance

of the LED sources simultaneously improves exponentially with temperature. Finally

183

in § 5.5, we concluded our discussion with a promising observation that the primary

reason for both the improvement of the source diode and the degradation of the de-

tector diode is their decreased resistance to current flow at low voltages. Furthermore

since the detector diode in such a pair must have a bandgap energy equal to or below

that of the source to absorb its emitted photons, the improvements on the source

side can effectively compensate for the detector's decreased shunt resistance to create

a spectroscopy system whose signal-to-noise ratio actually improves with increasing

temperature.

6.2 Further Scientific Questions

The work described in this thesis spans a range from the basic to the applied. Most

of the work described in Chapter 2 and Chapter 3 was aimed at characterizing and

understanding the thermodynamics of light-emitting diodes under sub-bandgap bias

conditions. The latter chapters focused primarily on the application of such devices in

systems with exogenous goals like transmitting information with high energy efficiency

or extracting spectral information from a fluid sample at high temperature.

Although our scientific work on thermo-electric pumping in LEDs enabled cer-

tain classes of applications, several interesting scientific questions remain. Broadly

speaking we categorize them into those related to the role of entropy and informa-

tion in photons and those related to their role in electronic degrees of freedom in

semiconductor devices.

6.2.1 Entropy and Information in Photons

In Chapter 2 we chose to represent the excitation of the outgoing photon field as

a thermal state with an effective temperature T*. With the LED at forward bias,

T* would be greater than the ambient temperature, leading to greater occupation of

184

these outgoing modes compared to background thermal radiation, and thus an out

flux of optical power. This is not the only valid way to describe the state of these

modes as they carry completely incoherent radiation away from the device. In fact, an

alternate description in which the temperature is fixed and the field is taken to have

nonzero chemical potential p is more common. The same occupation, and therefore

spectral intensity can be described either way:

1 1f = or f= . (6.2)

e kBT* -1 e kBT -1

In fact, since the density matrix of a thermal field is geometric, it is determined

entirely by the ratio of probabilities of the mode containing n and n + 1 photons.

Since either the inclusion of At > 0 and T* > T serve to parameterize this same ratio,

these two descriptions correspond to the same density matrix, and therefore represent

physically identical states:

p = In) (nI (1 - )," (6.3)n

wherehw ______

r = ekBTr or r e kBT . (6.4)

In Chapter 2, we elected to use the T* description because this quantity could be

used in the expressions for entropy and energy so as to result in familiar and intu-

itive expressions for the Carnot limit of a thermo-photonic heat pump. Both of these

descriptions, however, fail in the degenerate limit. As p -+ hw or as T* -+ 00, the

ratio of probabilities approaches one, and the expected occupation sees a non-physical

divergence. This situation corresponds to the electron-hole system approaching trans-

parency, where our many of our assumptions break down including the assumption

that our sample is "optically thick."

185

An interesting direction for further work may be to characterize LEDs at voltages

just below the band-gap to see when this description breaks down. At inversion,

the thermodynamic theory has little to say- the electrons and holes have a negative

temperature and so eliminating a pair generates entropy regardless of the final state

of the photonic system. However, when real diode lasers reach threshold, the current

noise in the electronic system can pass through to the photon field. Since this mecha-

nism for the introduction of disorder into the photon field is not accounted for in the

present theory, it is unclear to the author whether or not such a mechanism would

begin to dominate even below inversion. In essence this amounts to characterizing

the domain of validity of the theoretical picture presented in Chapter 2.

Measurements of the average intensity and the intensity autocorrelation could

reveal a better understanding of the breakdown of this theory, and ultimately the

practical limits of using conventional semiconductor diodes for photonic heat pump-

ing.

First, the average spectral intensity emerging from an LED close to transparency

could be compared against the relationship between V and L from the low-bias limit.

From Chapter 2, our prediction would be for a spectral intensity given by the Planck

formula but suppressed by a factor of the absorption through the active region at

this bias condition. Since this quantity approaches zero as the device approaches

transparency, our prediction can be expected to diverge from reality.

Second, experiments similar to the one carried out by Hanbury Brown and Twiss

in 1956 [99] can be used to characterize the degree of coherence of a photon source. If

the light emerging from an LED at low bias were subjected to such an experiment, the

intensity autocorrelation could be directly measured and compared against existing

models of current noise in diodes to determine if and when a noise source other than

the fundamental noise from recombination of electrons and holes carrying entropy

becomes dominant. Further theoretical study of the problem would be required to

186

properly connect these experimental results to measurements of photon entropy given

that the signal contains both equilibrium radiation and radiation driven by externally

supplied electrical work.

In addition to the use of quantum optical techniques to characterize the domain of

validity for the theoretical predictions from Chapter 2, the work in Chapter 4 raises

interesting questions about the limits of efficient photonic communication.

The experimental link described in § 4.3 immediately raises the question of efficient

communication when the source and detector are not at the same temperature. If a

high-temperature emitter is connected to a low-temperature detector by an optical

path that interacts with matter on both sides, energy will flow from hot to cold in

the form of a net flux of thermal radiation even when the channel is not in use.

Since it is possible to extract work from such a temperature difference, that work

could in principle be used to encode information on the field leaving the emitter. At

first glance, it seems this strategy could be used to consume less than kBT ln 2 per

transmitted bit, or even to net generate power during communication. However, it

is not immediately obvious how one would extract the exergy from the net photon

flux while also allowing the signal to be recovered at the detector. Furthermore, since

the energy efficiency limit established theoretically in § 4.2 presumes the existence

of a perfect detector (i.e. all uncertainty at the detector was due to the entropy in

the incoming photon field), that result may be more logical to interpret as a limit on

the energy required to transduce known information from the electrical input signal

onto the optical output of the finite-temperature source, regardless of the state of the

detector.

Such an interpretation of communication evokes images of a familiar model from

statistical mechanics known as Maxwell's Demon. The Maxwell Demon tries to open

and shut a boundary between two halves of a container of gas at equilibrium. His

goal is to generate a temperature difference by preferentially allowing the fast-moving

187

particles transmit from left to right while letting the slow-moving particles transmit

from right to left. If such a Demon could perform this operation with negligible

power consumption, the final state of the gas could be used to drive a heat engine

and extract work in violation of the Second Law of Thermodynamics.

In this situation, the Demon is encoding known information about the state of

the individual particles composing the gas into the degrees of freedom describing

the particles' motion. From these degrees of freedom, a heat engine is then able to

extract work. In order to comply with the Second Law, the amount of work required

for the encoding process must be greater than or equal to the amount which could

be extracted from the final state. In the same sense, the LED in the link from § 4.3

is attempting to encode known information into the outgoing photon field. Since

the final state of that outgoing photon field has a higher temperature T* > Tattice,

the requirement of consuming kBT In 2 per bit encoded could be seen as the Maxwell

Demon's analog for electrical-to-optical conversion.

This interpretation in turn raises a possibility of eventual practical importance.

The exergy in the photons which comprise the signal in our LED link could be used to

drive a photo-current at the detector side. If the detector were operating as a perfect

photo-voltaic, the power recovered could be used to drive the source or to physically

represent the electrical signal that emerges at the receiver. Such a link doesn't con-

sume any power in the traditional sense; rather it allows a physical signal to flow

from the electrical domain at the source into the optical domain for transmission,

then back into the electrical domain at the receiver.

We note that this conceptual configuration shares characteristics with more ab-

stract models for zero-net-power communication emphasized by Landauer [49] in re-

sponse to misunderstandings he ascribed to overgeneralization of his original analysis

leading to the kBT ln 2 limit.

188

6.2.2 Entropy and Information in Electrons

The results described in this thesis also raise interesting questions related to the flow

of entropy and information through electronic degrees of freedom (i.e. motion of

electrons and holes). In the basic description from § 2.1 of electron transport through

a double hetero-junction LED with bias voltage V < Egap/q we saw that the electrons

and holes absorbed entropy from the lattice during injection and released entropy into

another reservoir during recombination. As we saw in § 2.2, the electrons are in effect

a working fluid for the heat pump. That is, just as in the case of a macroscopic,

mechanical refrigerating heat pump, a closed thermodynamic sub-system internal to

the pump (typically a two-phase refrigerant fluid such as R-134A [100]) is used to

absorb entropy from the reservoir being cooled and eject entropy into the reservoir

being heated.

Furthermore, as we briefly discussed in § 2.6, if we step into the "frame" of an

electron as it passes through the device, the local environment follows some path in

the space of the relevant thermodynamic state variables (i.e. temperature, specific

entropy, electro-chemical potential, and number density). When the device is com-

bined with a source of work to form a closed circuit, the corresponding path returns

to its original position and forms a closed cycle.

For macroscopic, mechanical heat engines and heat pumps, the development of

new cycles like the Sterling and Brayton cycles led to significant practical improve-

ments. Today a turbine engine using a Brayton cycle can be 60-65% efficient, as

much as double that of an engine running a simpler diesel cycle [100]. Similarly, new

device-level designs of LEDs designed to operate as thermo-photonic heat pumps may

be able to mold the flow of electrons into new, improved thermodynamic cycles. In

this way, the basic building block of semiconductor physics, the p-n diode, can serve

a similar role for semiconductor engines as the single-piston reciprocating engine did

for mechanical engines almost two centuries ago.

189

EE

EE

V X

TS S

T,,t,,x(I-qV/E g)1

Titfc T*

Figure 6-1: Two representations of electron transport in a forward-biased light-

emitting diode, vertically aligned to emphasize the connections between the models.

At top is a familiar band diagram, which is essentially a statistical model because it

attempts to describe the state of the system in terms of micro-states. For example,the probability for occupancy of a particular conduction band state can be calculated

from the electron quasi-Fermi level and temperature at that point. At bottom is

a thermodynamic model for the same physical device. Here the variable T* refers

to the temperature seen by inter-band processes and Seiecton and Shole refer to the

per-particle entropy of the electrons and holes respectively. These quantities can be

used to calculate the per-particle heat and entropy fluxes in the electronic degrees of

freedom. Since the bottom figure does not indicate any specific values for the pres-

ence of accessible states or their average occupation (i.e. there are no bands or Fermi

levels drawn), we say the description is thermodynamic rather than statistical.

190

We note that developments along this line are already underway for related tech-

nologies. For thermo-electric heat pumps and heat engines, which are typically com-

posed primarily of semiconducting materials, new cycles leading to significant system-

level improvements have already been developed [101, 102]. Researchers working to

improve the efficiency of solar photo-voltaics have begun to measure their progress

in terms of the suppression of a series of free energy losses [103], and thermo-photo-

voltaics have been reported which utilize thermo-electric heat exchange at hetero-

junctions to enhance their open-circuit voltage [104].

Finally we end our description of the outlook for thermodynamic considerations

in electronics by noting that even circuits with multiple devices may be subject to

similar thermodynamic analyses. For example, a basic CMOS inverter (composed

of a single NMOS and a single PMOS transistor) within a network of logic gates

can be analyzed in this way. As electrons flow from ground up to Vdd, they traverse

four metal-semiconductor junctions and four semiconductor p-n junctions in series. In

doing so, they irreversibly generate on average ASinverter = ( CVa)/T of entropy each

time they transport the information about the charge state of the gate electrodes from

their input to their output. Here C is the input capacitance of the subsequent stage

and T is the ambient temperature surrounding the larger logic network. In essence,

the gate is generating ASinverter to perform a reversible transformation on a single bit

of information and move a copy of the result to its output. By applying a similar

procedure to other logic gates and ever larger networks of gates, we can ultimately

build accurate thermodynamic models of entire computing machines. These models

may provide insights into energy-efficient computation along the lines of adiabatic

computation [48, 105] and computational sprinting [106], or lead to new capabilities

related to thermal physics like physical random number generation [107].

191

6.3 Further Applied Directions

The results described in this thesis also point to several areas of applied work. The

first, most obvious direction is to use the design developed in § 2.5 to grow, fabricate,

and test an infrared LED.

Despite their wider bandgap, similar projects in the indium phosphide and gallium

arsenide material systems could also hold promise if properly designed for deep-sub-

bandgap operating voltages (i.e. Egap - qV > kBT). In particular, the strategy of

doping the active region could lead to LEDs with very high quantum efficiencies in this

regime due to the low defect densities achievable today. Although the power density

in this regime will suffer from the wider bandgap, detectors at these wavelengths

should also have lower noise equivalent power for the same reason. For applications

like communication and spectroscopy, where signal-to-noise ratio can be more relevant

than total power, this strategy would let us experimentally test certain fundamental

limits which we have thus far relied on idealized extrapolations to explore.

Furthermore the general study of LEDs made from wider bandgap semiconductors

such as InGaN could help us understand the physics of these devices at sub-bandgap

voltages. In theory many of the phenomena described in Chapter 2 rely primarily

on both carrier species being in the Boltzmann regime. Thus another interesting

direction would be to characterize visible devices at voltages which are on the order of

the bandgap energy but whose carriers remain in the Boltzmann regime. In particular,

the extent to which the connection between voltage and light intensity from § 3.4.3

remains valid for these devices could lead to new understanding of the temperature-

dependence of efficiency for wide bandgap emitters.

Another interesting direction is to use the thermal physics expressed in § 6.1 above

to re-engineer an LEDs thermal design. Since the efficiency of an LED at qV < Egap

can be an increasing function of temperature, thermal packaging which maximizes

operating temperature could improve efficiency. As we alluded to in the publication

192

reporting the first demonstration of above-unity efficiency electrical-to-optical power

conversion:

"[For LEDs at sub-bandgap operating voltages,] self-heating may of-

fer a convenient solution for sources with subunity [wall-plug efficiency].

Here, purposeful concentration of internally generated heat, such as in an

incandescent filament, should allow phonons to be recycled to thermally

pump the emitter."

SANTHANAM, GRAY, AND RAM

PHYSICAL REVIEw LETTERS, 2012

For reasons related to a patent application which awaits examination (Provisional US

Patent Application No. 61/684315 filed August 17, 2012; full US Patent Application

filed August 16, 2013), we limit our discussion here.

Another very interesting direction to consider is the use of modern nano-photonic

and plasmonic techniques to further enhance the quantum efficiency of LEDs. In

the transport models from § 2.1, the coefficient B parameterizes the rate of radia-

tive recombination (cm-3 s- 1 ) for a given concentration of electrons and holes. The

microscopic physics contained in this figure in most cases can be obtained rather

straightforwardly using Fermi's Golden Rule. In such a calculation, the transition

rate is proportional to the absolute square of the matrix element connecting the

initial and final states of the combined electron-photon system and is linearly propor-

tional to the density of final photon states available for emission. Using nano-scale

structures composed of dielectrics and metals, the local mode structure of the photon

field can be distorted to allow a particular region of space and optical frequency to

have a density of states much higher than in vacuum. Via the Purcell effect, the

rate of radiative recombination can therefore be enhanced, and with it the device's

quantum efficiency.

While other opto-electronic devices like visible LEDs ultimately seek to interact

193

with photon modes in vacuum, solid-state refrigerators using thermo-photonic effects

to pump heat from the lattice of one diode to that of another diode do not. As a

result, metamaterials with very high densities of states can be used to increase the

radiative thermal conductance, which determines the heat pump's power density. Re-

cent experiments have confirmed that the evanescent tails of photon modes can be

used to enhance radiative heat transfer while minimizing conductive and convective

heat transfer across nano-scale gaps [108]. By exploiting this principle to control

heat transfer across a narrow gap separating two diodes which use the Purcell effect

as described above, thermo-photonic heat pumping with high power density may be

achievable [109]. We note also that similar strategies for thermo-photo-voltaics have

recently come under consideration [110, 111] including those using metamaterials ca-

pable of drastically increasing radiative heat transfer in the near-field [112, 113]. Early

work suggests enhancements of two to three orders of magnitude may be realizable

[114, 112, 115].

In parallel with the development of improved sources, their employment in high-

temperature mid-infrared spectroscopy systems represents a near-term target to which

heat pumping LEDs could lead to real system-level benefits. For example, the use of

LEDs and photo-detectors in the InAsSb ternary alloy system can be used to perform

absorption spectroscopy around 3.4 pm. From the spectra in Figure 5-4, we saw that

if the optical path between source and detector includes an ATR crystal exposed to

a crude oil sample, spectroscopy at this wavelength could be used to fingerprint the

hydrocarbon chains present in it. Extrapolating from the exponential dependence of

LED efficiency and photo-diode shunt resistance with temperature would suggest such

a system would suffer from an extremely low signal-to-noise ratio in the photo-current

signal. However, making use of the source-detector compensation technique from § 5.5

should allow spectroscopic data around 3.4 ym to be acquired with reasonable lock-in

integration times.

194

Applied work on communication channels with very high source-side energy effi-

ciency could also be pursued. The magnitude of the dominant noise in the channel

presented in § 4.3 is likely quite close to the magnitude of the noise expected from the

random arrival of thermal blackbody photons. Thus relatively simple modifications

to the detector, such as further reducing its temperature or using a similar detec-

tor with a smaller absorptive area, could allow us to develop a channel whose noise

is dominated by fundamental sources connected to the temperature of the source

LED. In fact, the data presented in § 5.5 suggest this may be accomplished simply

by using the source-detector diode pair from the spectroscopy experiments around

3.4 pm. Furthermore, if devices possessing high quantum efficiency in the low-bias

regime V < kBT/q can be developed in the InP or GaAs material systems, near-IR

experiments using low-noise cryogenic photo-detectors or visible experiments using

photon-counting avalanche photo-diodes could be used to form interesting channels.

According to the theory in § 4.2, a channel which is whose signal is encoded by a

low-bias LED that is nearly free of irreversibility and whose noise is dominated by

thermal blackbody photons should come close enough to the kBT ln2 limit to help

address the outstanding question of whether classical communication at optical fre-

quencies faces a limit stricter than kBT ln(2) because discrete photons carry energy

hw > kBT [116].

6.4 Engineering Toward Second Law Bounds

In this thesis we have presented the first experimental verification that an LED can

emit optical power in excess of the electrical power used to drive it, a concept which

was first introduced theoretically in 1957 [31]. As with any energy-conversion tech-

nology, the ultimate limits for the efficient generation of white light are set by the

Second Law of Thermodynamics. Using the theoretical framework we developed in

195

Chapter 2 and supported empirically in Chapter 3, for any incoherent light field a

sufficiently ideal electrically-driven source can generate it with a wall-plug efficiency

above 100%. That is, the device will harvest ambient heat to provide a portion of the

power which drives the source in steady-state rather than consuming more electrical

power than it emits in optical power and releasing the remainder as waste heat.

Indoor Light AM1.5 High-Power LED die

TCarnot

0%1 25

W 75

50 min il for I=1

10-6 10-3 1 103 106White Light Intensity (W/m 2)

Figure 6-2: The theoretical thermodynamic efficiency limit for generating white light

from a thermo-electrically pumped LED as a function of light intensity. The solid

curve marked with circles denotes the maximum efficiency permitted by the Second

Law in a 300K ambient. The dotted curve marked with squares represents the mini-

mum quantum efficiency required by an LED (at the relevant sub-bandgap operating

voltage) to achieve white light generation at 100% wall-plug efficiency. Here "white"

is taken to have the relative spectral intensity of a 5800K blackbody between 380nm

and 780nm and no radiation outside that band; the absolute spectral intensity then

scales with the intensity on the plot's x-axis.

Although all incoherent light fields carry with them some entropy, they do not all

carry the same amount. Relatively dim fields, for instance, can carry much more en-

196

tropy than bright fields with the same spectrum. Likewise, broadband fields can carry

more entropy than narrow-band fields with the same optical power per unit area. As

seen in Figure 6-2, the light emerging from a modern high-power LED chip (~ 106

W/m 2) faces more strict fundamental thermodynamic limits than the ambient light

present in an OSHA-compliant workplace (~ 1 W/m 2 ) [1171. We may also interpret

this result in terms more relevant for the thermal engineering of waste heat manage-

ment: in order to rid ourselves of the waste heat problem by achieving a wall-plug

efficiency of 100%, the bright LED die will require at minimum a quantum efficiency

of 90% in comparison to 75% for a source matched to the intensity appropriate for

human consumption.

By generating light for human use at a brightness higher than it will ultimately

be consumed at, we are paying for more coherence than we can make use of. In

an indoor lighting context, light from the bright high-power LED chip ultimately

undergoes entirely avoidable irreversible entropy generation as it scatters off diffusing

surfaces or being partly absorbed by dark surfaces, or else is finally consumed by a

human retina which is insensitive to its remaining coherence. By contrast, if the same

total optical power is delivered to the room but is generated over a large area, the

emitted photons are already maximally disordered. Moving to a wider area emitter

at constant power effectively removes the irreversible entropy generation step and

replaces it with entropy transport from the ambient environment. The decision to

continue engineering improved small, high-power LEDs instead of more efficient by

larger area panels is then essentially an economic one.

An economic analysis of the ongoing efforts to replace old, highly inefficient light

sources with efficient LED lighting falls outside the scope of this work. Nevertheless,

we choose to briefly present an argument for the long-term relevance of the preceding

conclusion. There exists an empirical law called Haitz's Law, modeled after Moore's

law for transistor density in integrated circuits, for the development of cheaper and

197

brighter LEDs. Haitz's Law states that the price per lumen of LEDs is cut in half every

28 months, and has held true for the last 40 years [118]. In recent years, the trend has

actually accelerated (in part due to market interventions by governments). Between

2007 and 2012, the price fell by nearly a factor of 10 to less than half of a cent per

lumen [119]. Meanwhile, the real (i.e. inflation-adjusted) cost of electricity required

to power these LEDs has remained virtually flat. As a result, the lifetime ownership

cost of is increasingly dependent on the operating expenses (power consumption) and

less dependent on initial capital costs. Thus, in the long run, a technology which uses

an expensive wide-area emitter but consumes less power could prove more economical

than a small bulb that produces requires less of the presently-expensive semiconductor

area per lumen of lighting capacity.

Since the earliest surviving formulations of the Second Law of Thermodynamics,

those of Lazare and Sadi Carnot in 1824, it has been used to calculate the fundamental

limits of machines whose primary purpose was energy conversion. The generality of

the constraints it applies to physical systems has allowed it to remain as relevant to

modern machines like LEDs as it was to steam-driven turbines in the 19th Century.

Following the early work of Maxwell and Boltzmann on the kinetic theory of gases,

a few decades later the notion of entropy began a parallel development in which

the Second Law could be formulated as a statement about information. In modern

statistical mechanics, instead of interpreting entropy as a substance attached to some

forms of energy which cannot be destroyed, we can interpret entropy as a measure of

unknown information which cannot be deleted. In the same way that thermodynamic

machines seek to mold the flow of energy for practical purposes but remain constrained

by the inability to destroy entropy, today's information processing machines seek to

mold the flow of information but remain constrained by the Second Law.

In Chapter 4 we discussed one of the simplest information processing machines: a

communication link. The goal of this machine is simply to transport information from

198

one physical system to another. Because information is physical, we found we were

unable to encode the information we wanted to transport without imposing some order

on some physical subsystem, in this case an interval of photonic phase space, that then

travels from transmitter to receiver. This order, or equivalently known information

about the physical state of this subsystem, required a certain nonzero amount of

energy to be added to the subsystem because imposing this order without moving

any energy between subsystems would constitute a violation of the Second Law in

that subsystem. In this sense, the kBT In 2 per bit limit for our channel's efficiency

is a consequence of the Second Law. The Landauer limit is not a statement about

communication, but a statement about the representation of known information in

physical systems. It applies equally to any physically realizable information processing

machine.

In short, all descriptions of physical state contain information and all real infor-

mation must be represented physically. In the long run, we will inevitably invent

new technologies, but the constraint of the Second Law on all information processing

machines will remain. As a result we should expect our information technologies,

such as those which perform digital communication, to follow the same trajectory as

energy conversion machines in their inevitable march toward the bounds set by the

Second Law.

199


200

Appendix A

Entropy and Temperature of Light

The basic result about classical LED communication which this project seeks to

express relies on a thermodynamic analysis of the low-biased LED. Without a proper

understanding of entropy and the effective temperature of light, we cannot consider

the electronic subsystem of an LED to be the working fluid of a heat engine operating

between one thermodynamic reservoir of phonons and another reservoir of photons,

and so cannot derive a Carnot bound for the efficient generation of thermal photons.

The papers summarized in this document, therefore, constitute an important part of

the literature supporting the communication result.

The process of defining a temperature for use in a thermodynamic analysis of an

incoherent light-emitting device has three basic steps.

Step One: Dividing Up Phase Space

The first step in analyzing the thermodynamic properties of light is to find chunks

of phase space which can be treated as individual quasi-equilibrium systems. To do

so, we look for intervals in 6-dimensional phase space in which the average photon

occupancy is roughly constant. The 6 dimensional interval is the intersection of an

interval in three spatial dimensions and one in three reciprocal-space dimensions;

201

the latter could be equivalently described by a frequency interval and an interval of

solid-angle for the propagation direction. Intervals within which the light intensity

is directly proportional to the photon density-of-states may be said to have constant

occupancy.

Within such an interval, thermodynamic state-variables for the photonic system

may be calculated from a single mode with the values for extensive state-variables

scaled with the number of modes. In particular, for incoherent thermal light the

average occupancy is sufficient information to know the entire state of a mode for

reasons outlined in Step Two. Making things even more convenient for thermal light,

the average occupancy may be simply calculated in any situation as the average

energy per h3 /2 of phase space, divided by hw.

In some publications which seek to calculate the effective temperature of the light

emitted in some specific situation, this breaking of phase space is the first step.

In 1959, while exploring the thermodynamic limits of efficiency for lamps, Wein-

stein [29] published basic calculations assigning an effective temperature to the light

from a green ZnS phosphor. In this calculation, he approximated the emitted light

with a gaussian emission spectrum of width Av around some center-frequency vo.

By doing so, Weinstein implicitly assumes that the only portion of phase-space of

relevance to the calculation is that around the primary fluorescence frequency. In

1980, Landsberg and Tonge [33] chose to treat analytically the case of the gaussian

spectrum modulated by a power-law and calculated the effective temperature in this

general case. It is assumed that the utility of this result lies in assuming that the pho-

ton density-of-states may be easily approximated by a power law within the relevant

frequency band, even in the case of Purcell or photonic crystal effects.

To simplify things further, some authors in fact choose to examine only one finite

interval of phase space within which they proceed with a statistical analysis, and

outside of which the photons are negligible.

202

For example, when Mungan [30] applies this basic framework to the first exper-

imental observation of net-cooling in a solid (Epstein et al [120]) for pedagogical

purposes, he makes similar assumptions about both the incoming pump laser light

and the fluorescence emitted by the Yb3 +:ZBLAN(P?) glass being cooled. In partic-

ular, he describes the laser as having constant intensity within the entire phase-space

interval of relevance. Although laser light need not be thermal, Mungan's assump-

tion that the entropy may be calculated in this way is justified by the conclusion he

reaches. Although Mungan does not explicitly address the issue of non-thermal pho-

ton populations, by showing that were the laser light thermal (a quantum-statistical

state with maximal entropy per unit energy) the entropy it would carry would still be

negligible and Taser -+ oc could be assumed in the subsequent thermodynamic anal-

ysis. In effect, the thermal-light assumption lower-bounds the effective temperature

of the laser light. For the case of the emitted fluorescence, Mungan performs two

separate calculations: first he assumes that the light is of constant intensity within

a bandwidth given by the full-width at half-maximum of the measured spectrum,

then he goes ahead with the full average-temperature calculation for the real mea-

sured spectrum. In the first (flat power-spectral density assumption) case, he finds

the outgoing radiation to have TF = 1760K; in the second case, he finds the outgoing

radiation to have TF = 1530K. This calculation suggests that for spectra typical of flu-

orescent ytterbium, the flat-power assumption results in effective temperatures with

a 10-20% error. The flat-band calculation over-estimates temperature because the

tails of the distribution are not included; if the effect of non-flatness within the band

were of dominant consequence, the effective temperature would be under-estimated.

This final observation relies on the concavity of entropy, to be discussed in Step 3.

On the other hand, Weinstein later showed [121] that the quantum-mechanical

inverse relationship between spontaneous-emission linewidth and emitting carrier life-

time is necessary to resolve the apparent breakage of the 2nd Law when a hot system

203

of oscillators relaxes by undergoing radiative transitions. The lesson appears to be

that while the effective temperature of light is typically not sensitive to the exact

shape of the spectral density, the characteristic linewidth for a given integrated in-

tensity is critical to satisfying the Laws of Thermodynamics.

Step Two: Drawing the Entropy Function

Electromagnetic modes should be thermally occupied if their occupancy results from

interacting with some composite of microscopic electronic subsystems with nonzero

matrix elements for photon emission/absorption (i.e. oscillators) whose entropy and

energy are related by a single temperature. In the case of an LED, for example, our

subsystems are pairs of vertically-aligned conduction and valence band states whose

upper-radiative (UR) state occupancies are given by the Fermi level and tempera-

ture of the electrons and whose lower-radiative (LR) state occupancies are likewise

given by the hole quantities. If the two temperatures and Fermi levels are the same,

then it is clear that whether we slice the E-k diagram horizontally (define bands as

subsystems) or vertically (define oscillators as subsystems), every quantum state is

occupied in equilibrium with every other state, so any photon modes that have come

to equilibrium with this system should also be thermally occupied at the same tem-

perature. An analysis of small deviations of the AEF- and AT-types should reveal

that as such light-emitters are infinitesimally turned on, the photon fields with which

they interact should be continuously deformed and the assumption of thermal light

should remain valid.

If an electromagnetic mode is said to be occupied with thermal light, then its

density matrix is diagonalized in the number basis and the ratio of probabilities

for occupancy by n + 1 photons to the probability for n photons is a fixed number

independent of n. We may think of this ratio, related to the temperature of the mode,

204

as fixed by the basic construction of the canonical ensemble from statistical mechanics.

In the canonical ensemble, our single photonic mode interacts with a reservoir whose

entropy (log of number of configurations) increases by the same amount with the

addition of each unit of energy, regardless of how much energy has been pulled from

or put into that reservoir by our mode. Because the ratio of probabilities is fixed,

the occupancy probability distribution (diagonal elements of the density matrix) is

geometric and so self-similar. The self-similar nature of this probability mass function

(PMF) leads to a simple recursion-relation to calculate its entropy:

H(P) = Hb(r) + rH(P) where r = (n±1)P(ri)

r log(r) + (1 - r) log(1 - r)H(P) = -1ro()+ 1r

rand since N = (f)p = then

- r (A.1)

H(P) = -1 Nlog + log( )I N + I N + 1

= (N + 1) log(N + 1) - N log N

-+ S(N) = kB [(N + 1) log(N + 1) - N log N]

The entropy function is plotted as a function of occupancy N in A-1 and as a

function of the probability-ratio r in A-2.

While the basic mathematical structure of the preceding derivation relies on Bose

statistics, the actual formula appears to have been derived in several different physical

models since Planck [122, 123, 124, 125, 126]. In some cases, the authors have chosen

to treat the photons as a closed statistical system of bosons [123, 124, 125], as we

have in the preceding discussion. In other cases, the authors have chosen to define the

entropy of the photon field by the temperature of a coupled reservoir of electromag-

netic oscillators which has exchanged energy and entropy to reach a detailed-balance

equilibrium state with the photons [122, 125], as in the canonical ensemble. Since

205

4

.C1

CL

0 2 4 6 8 10Occupancy

Figure A-1: Entropy of a thermally-occupied photon mode (blue line) as a functionof expected occupancy. Since the expectation value for the energy is just Nhw, whichis just a rescaling of the horizontal axis, the qualitative behavior of this functionS(N) is the same as S(U). The thick red line indicates an approximation to thisformula which comes from considering only the binary variable indicating whether ornot the first photon is present. Note that for average occupancies < 1, this Fermion-like entropy function approximates the full entropy. Successively thinner red linesindicate inclusion of the second and third photons' presence or absence.

206

4

c\I 4C3

4

CL o

0 0.2 0.4 0.6 0.8 11 - Probability(GroundState) = Probability(AnyExcitations)

Figure A-2: Entropy of a thermally-occupied photon mode (blue line) as a functionof the characteristic ratio r = P(n + 1)/P(n). The red lines approximate the fullBosonic solution with simple binary random variables including the first, second, andthird photons. Note that the first photon's entropy is just the entropy of a binary-rrandom variable, as we'd expect for a Fermionic mode occupied with probability r.

the statistical results for these oscillators are then derived microscopically by treat-

ing them as bosons, the result is identical. Finally, the most accessible derivation

of the entropy for a photon field comes from simply inverting the blackbody energy-

density and using the 3rd Law of Thermodynamics to recover the entropy expression

[125, 126]. Although this formulation hides all of the quantum mechanics behind the

Planck blackbody formula, it is reproduced here because of its simplicity.

Starting with the expression for the energy density U of a blackbody within a

frequency band Aw, we have:

207

44

b

3

U(T) = Aw x (density of modes at w) x (# photons per mode at w) x

h (e 2 B3 )7r20 exp hw/kBT - I)

=- T- 1(U) =kB

noIn hW3A+1 as

I T2c3U aU

Now, since by the 3rd Law of Thermodynamics, S -+ 0 as U, T -+ 0, we can inte-

grate dS = dU/T to find the entropy-density S at finite temperature (and therefore

energy-density). Defining the dimensionless energy-density as

72C3A = U F (A.3)

we have

I dS =S= U UkB

hwkB

hw

kB

hwkB

hw

dU'hw

hW3Aw72C3

hw3AW

hW3AW

723

hW3,Ao

7203

hw3AWi r2 c3 U'

dii' ln(1

+ 1'

+ V) - ('

v=1+ii

(vIn vvI =1 (fi t ,0)

{(ft+ 1)ln(ii + 1) - (ii+1) - IlnI+1-dnii+f±+0 -0}

{(i + 1) ln(t +1) - iln f}

(A.4)

With the entropy formula in hand, we examine the relevant concepts of tempera-

ture for light.

208

( photon)

(A.2)

Step Three: Defining Temperatures

Using the entropy function from the previous section, two different notions of temper-

ature are commonly defined when examining the thermodynamic limits on particular

photonic devices [29, 33, 30].

First is the brightness temperature,

= S (s-1 -o ITB l (1+k) (A.5)

aU kB 10g (1 + )O

which intuitively generalizes the notion of temperature from the micro-canonical

ensemble of closed photon gas-systems. Some authors, including Landau in 1946,

appear to have chosen to work only in terms of this temperature [123, 125, 126]

presumably because of its simple relationship to the Planck formula for blackbody

light intensity.

Second, there is the flux temperature

U hwNTN (A.6)TF ~ k (N + 1) log(N + 1) - N log N

which is useful for direct use in thermodynamic formulations of light-emitters whose

outgoing photon fields are far from equilibrium with their incident fields. Since the

chunks of phase-space that we broke our problem into in Step One are continuously

streaming at the speed of light in configuration-space while our physical apparatus

remains stationary, we are often faced with the question of how much total energy

and entropy has left or entered our device with a given pulse. In this case, we care

about the entropy contributed by each and every photon, not just the last photon

that was added to the pulse. For this reason, the quantity with units of temperature

(energy/entropy) which determines the entropy flux associated with a unbalanced,

unidirectional photon energy flux is defined as TF.

209

Now let's examine the relationship between TB and TF. Since knowing TB deter-

mines the thermal occupancy N for a mode with given w, and N can in turn define S

and thereby TF, an explicit analytical relationship between TF and TB can be found

directly by substitution. By defining XB B and XF k' to be dimensionlessykBTB kB TF

inverse temperatures, we have

XF = ( 1) log(N + 1) - log(N)N

exp XB expxB -1 expB g 1- -log -log

expXB-l 1 expX B - 1expXB- / (A.7)

= exp XB log(exp XB) + (1 - exp XB) log (exp XB - 1)

= exp XBXB + (1 - exp XB) log (exp XB - 1)

In the low-intensity (i.e. low-TFB, high-w, high-xFB) regime, the log can be

approximated by its Taylor expansion, resulting in the simplified relationship

XF eXp XBXB + (1 - exp XB) (XB - exp -XB)

= XB - exp -XB +1 (A.8)

~XB +l

so that in this limit, the notions of temperature converge (TF -4 TB) as

TF=TB 1+ kB< (A.9)

but that TF always remains below TB. This final fact remains true even outside of the

low-occupancy limit and is a consequence of the concavity of entropy to be discussed

shortly. All of these results regarding the relationship between TF and TB appear

where they are useful throughout the literature [29, 33, 30].

One last point to focus on is the importance of the concavity of the entropy

function. Although it may be proven rigorously that the entropy function of any

210

probability distribution is concave (i.e. the entropy of any mixture of variables is

necessarily more than the sum of the entropies of the variables alone; the choice of

which variable to use itself contains entropy) here we only note that the statement is

true for the family of thermal photon-occupancy distributions. The concavity of our

distribution can be seen visually in A-1: the line connecting any two points on the

curve lies entirely below the function S(N).

As we mentioned before, concavity shows us that the brightness temperature TB

(inverse slope of tangent-line to S(N)) is always greater than the flux temperature

TF (inverse slope of line from the origin thru S(N)). That is, each additional photon

that stacks up in a given volume of phase-space contributes less to the entropy than

the one before it. Additional power always brings additional entropy flux, but also

always suffers from a law of diminishing returns.

Concavity also helps us recover our intuition about linewidth and entropy. To see

this, consider two similar physical situations. In the first, M-many photons uniformly

occupy all of the states within a frequency range from wo to wo + Aw; all photon

modes outside this frequency interval are empty. In the second, let the M-many

photons instead uniformly occupy all of the states within a frequency range from WO

to wo + 2Aw. To find the relative amounts of entropy contributed by the photons in

each situation, we can simply notice that each mode that matters in the second case

carries exactly half as many photons as each in the first case. Since the entropy of

a half-as-occupied mode is necessarily more than half that of a fully-occupied mode,

the entropy in the second situation is necessarily greater. Running the argument in

reverse, as as we decrease the linewidth (or any dimension of the phase-space for that

matter, including directionality or volume) but maintain the same number of photons,

the entropy-per-energy tends to zero. Consequently, the effective temperatures TF and

TB both diverge and the entropy of such light has no consequence thermodynamically-

the light energy might as well be work.

211

Photonic irreversibility, a topic explored by Planck nearly a century ago [127],

can likewise be seen from this perspective. The motion of electromagnetic fields in

any real situation are also constantly seeking the local maximization of entropy. For

thermal light, this is equivalent to attempting to spread out in phase-space, with

photons preferring less-occupied modes over highly-occupied ones, where they can

contribute a greater amount to the total entropy.

Interestingly, this behavior is the exact opposite of what photons experience in a

laser, where stimulated emission bunches photons into well-occupied modes. To lase,

however, the local photon field must interact with a population of inverted systems,

for which the release of energy is accompanied by an increased entropy. Since we

did not include this non-photonic entropy in our earlier analysis, the situation with

the laser does not break the 2nd Law nor our intuition for irreversible processes in

purely-photonic systems in which each mode is populated by thermal light.

212

Appendix B

Maximum Efficiency at 1 Sun

For the purposes of numerous applications involving biological organisms, the inten-

sity of solar radiation at the Earth's surface is of particular interest. Although the

sun's radiation is well-approximated as coming from a 5700K blackbody, its intensity

diminishes as it spreads out from the surface of the sun (Area=47rR') to a spherical

shell with radius given by the Earth's orbit Rorbit (Area=47Rorbit). The radiation

simultaneously becomes more collimated, thereby preserving the phase-space density

of photons and avoiding the associated entropy increase.

On the other hand, light-induced biological processes are frequently described in

terms that refer only to the longitudinal momentum distribution (i.e. power per unit

area per unit frequency) and ignore the angular distribution of the incoming light.

For these processes, the incident-angle-averaged power spectral density Io(f; T) of

the incoming solar radiation is apparently the relevant quantity. If the quantity of

interest is in fact Io(f; T), then permitting the angular distribution of a given flux

(such as that from the sun) to be more spread out (such as in an LED emitting at

1 sun in a given band) is equivalent to allowing the photons to explore more phase

space and carry more entropy. Thus the Carnot limit for the efficient generation of

such a flux should be looser than simply q 5 Qcarnot = Tsun/(Tsun - Tambient). Instead,

213

the limit for the angularly spread-out light should be characterized by the brightness

temperature of that radiation TB(f).

Our goal here is to find the Carnot limit for the efficient generation of "1 sun" of

incoherent radiation, as a function of frequency.

We begin by using the Planck blackbody formula for the frequency-dependence of

the light intensity at the surface of the sun:

1Isolar-surface(f) = Io -_- _ . (B.1)

expkBTsun -1

At the surface of the earth, where the term "1 sun" is defined, we have:

I1 1-sun = Isolar-surface (f) - , (B.2)

where G is a geometrical factor that describes the degree of collimation of light from

the sun:

G = Ror)it 2 ~46000. (B.3)(Rsun

At each frequency w, we may define the brightness temperature TB(w) of the

incoming radiation as the temperature of blackbody whose angle-averaged power

spectral density matches I1isun(w).

IB (W) = - sns (B.4)

1 1/Ge - 1 (B.5)

exp keB -1 eXp kBTsun-

expkBi ( = exp c$run -) + 1 (B.6)

kBTB = (B.7)In [G (expkarun -) +

(7

From here, we may employ the well-known formula for the Carnot efficiency for

pumping heat from a Tambient ambient up to the brightness temperature TB at each

214

TB(f)77(f) < '7Camnot(f) =T

TB(f) - Tambient

UI

5000

4000

3000

20001

10001

300K AmbientJ500 1000 1500 2000 2500

Wavelength (nm)

25

20C

15

10fli0

0

0

0

0

500 1000 1500 2000Wavelength (nm)

Figure B-1: Plots demonstrating the thermodyn~Tam~%"Ie vance of the distinction

between collimated solar light (red) and angularly-diffuse light of the same power

spectral density (green) at "1 sun" intensity. Left: Brightness temperature as a

function of photon wavelength (plot corresponds to analytical result in (B.4)). Right:Corresponding Carnot efficiency (computed using (B.8)).

These analytical results are explicitly plotted as a function of photon wavelength

in Figure B-1, where we have assumed Tambient = 300K for the efficiency calculations.

Averaged over the 5700K blackbody spectrum, the maximum efficiency for generation

of angularly-diffuse "1 sun" light is ~ 129%.

215

frequency:

(B.8)

7U

CL

a)

F-

U)

0 -

0

C2500


216

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